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UNDERSTANDING
FORCES AND MOMENTS2
For some reason, the terms forces and momentsdo not always
seem to be thoroughly understood. It is true that the English
language seems to suffer over a period of time, but in the area of
mechanics it is important to understand exactly what each term
means and to use these terms properly. The terminology whichfollows will be used in a practical manner. There are exacting
definitions that may be confusing to many while there are
descriptions that may convey a practical meaning to most
clinicians.
Orthodontic clinicians know from
personal experience that a specific
force system does not necessarilyproduce the same response for
different patients. Nothing in life
happens without a reason. Force
magnitude can be very significant.
as stated in Figure 2-1.
Figure 2-1
With an intrusion arch molars might erupt and/or incisors may
intrude. Bicuspids are more likely to undergo an equal and
opposite rotational response with powerchain elastics. These
responses are illustrated in Figures 2-2 thru 2-4.
FORCE SYSTEMS
Thesame force system may produce a
variable response.
Force magnitude is a significant factor.
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Molars may
erupt.
Forces may produce
variable response.
Forces may produce
variable response.
Incisors may
intrude
Teeth may rotate
Forces may produce
variable response.
Figure 2-2 Figure 2-3 Figure 2-4
The following illustrations may help to clarify some of the
misconceptions that are present in the orthodontic profession.
TRANSLATION
When a force acts through the Center of
Resistance or Center of Mass, only bodily
movement takes place.
FORCES (MxA)
Forces act in astraight line.
Forces consist of apush or pull.
Figure 2-5 Figure 2-6
In Figure 2-5, a force is applied through the center of mass, aterm used in reference to a free body such as a golf ball or
baseball. When the same force is applied through the center of
an attached body - such as a tooth - the term used is center of
resistance. This is nothing new to the orthodontist, but building
blocks will slowly be established so that confusion does not
arise later when discussing biomechanics.
The definition of a force could properly be defined as MxA
(Mass times Acceleration), but what meaning would this have
for the clinical orthodontist? If we describerather than definea
force, it can be seen in Figure 2-6 that a forceacts in a straight
line and may consist of a push or pull.
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Retracting cuspids with an open coil spring does not result in
forces acting in a curve. If wepushfrom the lingual surface of a
tooth with a lingual arch, or pull from the buccal surface of a
tooth with an archwire, the force acts in a straight line as itpasses through the tooth.
Figure 2-7 demonstrates this by using descriptions rather than
definitions which so often confuse the issue. Depending on
exactly where these forces act, moments may or may not be
produced. This will be discussed later during the subject of
forces and moments.
Push fromthe lingual
Pullfromthe buccal
Forces act in
astraight line
not a curve1
2 3
Figure 2-7
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MOMENTS (FxD)
Moments are produced as a
result of forces actingaway
from the Center of Resistanceor Center of Mass.
ROTATIONThe product offorce x distance produces
the moment on the body.
Therefore, 1/2 the force x twice thedistance produces the same moment as
1/2 the distance x twice the force.
Figure 2-8 Figure 2-9
When a force acts on a body, but away from the center of
resistance(or center of mass), there is a perpendicular distanceestablished between the applied force and the center of the
object as shown in Figure 2-8. It is the product of this distance
and the force that produces a moment. In other words, if either
the force or the distance doubles, the moment produced would
double. This is significant because in Figure 2-9 it can be seen
that different force magnitudes can produce the same moment.
If one force is half the magnitude of the other, but acting attwice the distance, the moments in each case will be equal. This
is important to recognize in orthodontic treatment as it affords
the opportunity to produce desirable moments without the
disadvantage of high force magnitudes, particularly in the
vertical plane of space where vertical dimension of the patient
might be compromised.
Personal experience in our lives can be of great help in
recognizing forces and moments produced in orthodontic tooth
movement. Most of us have probably played the game ofpool-
often referred to as billiards -sometime in our lives or at least
observed it being played by others. It is quite popular on TV.
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So, lets take a look at the game
of pool and see how it may be
of help in learning. For those
who may not be aware, the ballin question is known as the cue-
ball and is the white one seen in
Figure 2-10. Keep this in mind
so as not to become confused
Figure 2-10 with conservation of momentum
which involves the other balls. The following examples are
those we have experienced or can experience in our daily lives.
Visualize thecrown of
a tooth as acue-ball
Cue Stick
This represents the Pointof Force Application
The Cue Stick represents the
force that will be applied to the
brackets and tubes of the teeth.
Figure 2-11 Figure 2-12
The first step involved is to visualize the crown of the tooth as a
cue-ball as seen in Figure 2-11. The next step will be to identify
the point of force applicationshown in Figure 2-12 . The cue
stick used in the game of pool will represent the source of the
applied force. The next question is: In what direction will the
cue-ball move and how will it rotate? Keep in mind that the
rotation will be clockwise or counterclockwise in pool this is
referred to as left or rightEnglish. Naturally, the ball will roll
down the table due to friction, but disregard this rotation.
For those
who are
unaware,
the cue-ball
is the white
ball.
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CUE-BALL CONCEPT
1. A force applied through thecenter of abody will cause the body to move in astraight line and in the same direction
as the applied force.
2. A force appliedaway from the center ofa body will cause the body to move insame previous direction, but rotation willalso occur as a result of the momentcreated by the line of force acting at aperpendicular distance to the center ofthe body.
Figure 2-13 Figure 2-14
There are three possible
movements that may occur, just
as in the real world oforthodontics. The first movement
we observe is pure translation as
seen in Figure 2-13. The force
has been applied through the
center of the bodies shown.
Figure 2-15
Translation and rotation may occur as shown in Figure 2-14where the force has been applied away from the center of the
body as illustrated. The moment in such a case is referred to as
the moment of a force.
Figure 2-15 shows equal and opposite forces (known as a
couple) being applied and producing pure rotation.. The moment
in such a case is referred to as the moment of a couple. A pure
moment always acts around the center of resistance. Regardlessof the where the equal and opposite forces are applied, the body
will undergo pure rotation around the center of resistance.
Lets see where this concept applies at the clinical level.
3. Equal and opposite forces applied on a
body in the same plane of space and
parallel to each other (Couple) will
produce a pure moment causing the body
to rotate only.
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Translation
The result of the
applied force is a
moment and a
force at the centerof resistance.
Center bend
producing equal
& opposite
moments to those
already present. This is anEquivalent Force System.
Figure 2-16
In Figure 2-16 upper left, forces have been applied at the crownlevel resulting in tipping moments. The force system is always
shown at the center of resistance. Remember that a force
applied away from the center of a body will cause the body to
move in the direction of the applied force and rotate because of
the perpendicular distance. With the addition of a center (gable)
bend shown in the lower part of the illustration, moments
opposite to the tipping moments are created thereby eliminating
tipping moments measured at the center of resistance.The resultis that only pure forces remain as seen on the right in Figure 2-
16. This is referred to as an equivalent force system. Remember
the so-called powerarms that were introduced to the profession
in order to create a translatory force through the center of
resistance? Where are they now? Does this tell you how
successful or unsuccessful the results have been?
A clinical example of the above application is seen in Figure 2-
17. Tipping moments are eliminated by equal and opposite
moments resulting from a center bend. As will be explained
later, all archwire bends are done intraorally and activated 45
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degrees. All adjustments for
increasing or decreasing
moments for the proper force
system are created byadjusting whatever closing
mechanism is in use, such as
coil springs or powerchain.
In Figure 2-18, tipping the
Figure 2-17 incisors together would not
be acceptable. The placement
of a center bend into the wireproduces moments which
then result in bodily
movement as a result of
eliminating the tipping
moments produced by the
closing mechanism which
could be coil springs orpowerchain elastics.
Figure 2-18
Translation and Rotation
Figures 2-19 and 2-20 demonstrate that a force applied away
from the center of a body will cause the body to translate and
rotate. Looking at a rotated bicuspid with space mesial to the
tooth, it can be seen that applying a mesial force at the bicuspid
bracket will produce the necessary force and moment. This
obviously simple approach is intended only to illustrate the cue-
ball concept regarding translation and rotation.
A closing force at the brackets
producestipping moments
eliminated by Center Bends.
Tipping Moments
Eliminating the
Tipping Moments
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Figure 2-19 Figure 2-20
C.C.
Translation & Rotation Required
Translation & Rotation Required
Figure 2-21 Figure 2-22
Figure 2-21 and Figure 2-22 demonstrate the same concept
beautifully as will be seen later when wire/bracket relationshipsare discussed. For now however, simply keep in mind that by
excluding the second bicuspid brackets from the archwire, an
off-center bend has been created without the need to remove the
wire. In a full appliance the toe-in bend at the molar would
actually be a center bend when related to the adjacent molar tube
and bicuspid bracket on each side. By not engaging the wire
into the second bicuspid bracket an off-center bend has beencreated. Do you remember the rules for off-center bends? An
off-center bend contains a long and a short section. The short
section points opposite to the forceproduced thereby indicating
a buccal force on the molar. The toe-in bend (short section) also
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produces a rotational moment. This approach allows both
correction of the molar rotations and crossbites simultaneously
without removal of the archwire or use of crossbite elastics.
This is only one of many similar approaches that can minimizechairside time for the orthodontist as well as providing a variety
of noncompliant and exciting approaches not taught in school.
This might be a good time to mention that in over 46 years of
practice - thus far - never has the author used a crossbite elastic,
transpalatal arch, lingual arch, or any other type of lingual
attachments. Why not? Because there are so many alternative
and noncompliant approaches that do not require this. Manyother types of laboratory appliances which are commonly used
today can also be avoided. This will be discussed in the
upcoming chapters.
Pure Rotation
The final cue-ball concept relating to pure rotation - moment of acouple - can now be illustrated. Remember that equal and
opposite forces produce a couple.
Surgical
Exposure
Moment of a couple(Pure Rotation)
Figure 2-23 Figure 2-24
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shows that intrusive forces may cause buccal displacement of
molars due to buccal crown moments produced.
Intrusive force
producesbuccal
crown moment.
Extrusive force
produces lingual
crown moment.
IMPORTANT!
Intrusive force
producesbuccal
crown moment.
As the upper molars are widened,the
curve of Monson increases and no longer
harmonizes with the curve of Wilson.
Figure 2-28 Figure 2-29
In the lower part of the same illustration, it is seen that an
eruptive force acting through the molar tube produces exactly
the opposite moment and therefore possible lingual
displacement of molars. These undesirable responses may or
may not occur. Steep cusps and brachycephalic individuals with
strong musculature are only some of the factors which may playa role. When such undesirable movements do occur, an easy
solution is provided by the utilization of molar control bendsto
be discussed later.
In Figure 2-29, it can be seen that buccal displacement of the
molars may also result in an increase in the curve of Monson
an important functional curve involved in axial loading. It isthis type of occurrence that contributes so much to instability
and the increase in permanent retention seen today.
Since functional curves are an important part of orthodontic
treatment, this topic will be discussed now.
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Functional Curves
Figure 2-30 Figure 2-31
Three important functional curves are shown in Figure 2-30. In
Figure 2-31, it can be clearly seen that excellent axial loading is
achieved in #1, as the curves of Monson and Wilson nicely
coincide. However, in #2 there is an excessive curve of Monson
while in #3 there is a reverse curve of Wilson. In the latter two
cases there is a loss of axial loading which is apparent. These
discrepancies can very easily result from vertical forces acting
through the molars tubes as shown earlier. It has been shownthat eruptive forces through molar tubes create lingual crown
moments while intrusive forces acting through molar tubes
result in buccal crown moments.
The following illustrations will show the potential buccal and
lingual displacements that may occur as a result of vertical
forces acting through the molar tubes. If the second molars havenot yet erupted and the first molars are displaced without the
orthodontist being aware of such displacement, then upon
second molar eruption it may mistakenly be assumed that
second molars are at fault. As a result, treating to the first molar
width may then result in a faulty curve of Monson or Wilson.
The long axis should
lieparallel to the
Internal Pterygoidresulting inaxial
loading (stability).
Curve of Monson
2
1
Curve of Wilson
3
These curves
can be helpfulin determining
which arch is
involved and to
what degree.
Spee
Monson
Wilson
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Figure 2-32 Figure 2-33
In Figure 2-32 #1, the eruptiveforce has caused the first
molars to move lingually as
observed in #2. In #3, the
second molars have now
erupted. It remains important
to know which of the molars
are out of position. In Figure
2-33 the same series of eventsFigure 2-34 has occurred with first molars
moving buccally due to intrusive forces acting through the
molars. It can be observed that second molar eruption may
create the illusion that they have erupted too far to the lingual. In
Figure 2-34 it can be seen that casual observation could easily
lead one to believe the first molars are normal in width with
second molars being the problem.
The above movements make it important for the clinician to
include the functional curves of Monson and Wilson in
observing treatment progress. A failure to harmonize these
Buccal
Crown
Displacement
Original Molar Width
Change in Molar Width
2nd Molar
Width is
Normal
1
2 3
Lingual
Crown
Displacement
1
3
Original Molar Width
Change in Molar Width
2nd Molar
Width is
Normal
2
Second Molars are in normal transverse dimension.
Second Molars are in
normal position.
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curves may result in functional problems involving axial loading
and leading to later instability.
This concludes the chapter regardingforces and moments. Whatmay have appeared to be quite elementary at this point will
prove to be highly important in applying fundamental mechanics
in everyday treatment.
Most of what is contained in this book has not been taught as
part of an orthodontic curriculum. By understanding the
contents presented there will be many opportunities to treat
patients in a unique manner regarding the applied mechanics. Inaddition it will be discovered that there are many approaches
available that will lessen the need for patient cooperation
without the need for appliances that displace lower incisors
because of the undesirable reciprocal effects when treating
opposing arches with interarch appliances. You are about to
discover many ways of providing intra-archsolutions for many
malocclusions that will help to avoid placing appliances onopposing arches which may be normal and require no change.
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THE SHORT STORY
Forces and moments have been discussed in a manner somewhat
different than presented in the usual literature. Rather than
defining forces, they have been described. Description has
meaning to the practicing orthodontist whereas definitions
sometime seem to separate the academic nature of mechanics
from the reality of application to the patient.
It has been pointed out that force systems for the patient producevariable responses. Molars may erupt for one patient but not
another simply because of force magnitude. Other movements
such as reciprocal first and second bicuspid rotations tend to be
quite similar. It has been stressed that force systems must be
predicted and understood in order to effectively utilize them for
patient treatment. For many, biomechanics may seem like an
academic adventure because of unexpected responses. Differenttypes of responses have been demonstrated with the so-called
cue-ball concept and clinical examples illustrated. If the
orthodontist can begin to associate tooth movement with what
has been experienced in life, such association may gradually
lead to applications in orthodontic treatment.
Finally, the functional curves of occlusion have been presented.
It has been shown that the curves of Monson and Wilson should
be harmonized for axial loading during occlusion. Such
harmony contributes to stability. The forces causing a lack of
harmony have been presented and the clinician made aware of
their importance in observing functional curves.
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SUGGESTED READINGS
Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod
1984;85:294-307.
Dawson PE. Evaluation, diagnosis, and treatment of occlusal problems.
St. Louis: CV Mosby, 1989;85-91.
Mulligan TF. Common sense mechnics. 2. Forces and moments. J Clin
Orthod 1979;13:676-683.