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```Mechanics
of tooth movement
Richard J. Smith, D.M.D., Ph.D.,* and Charles J. Burstone, D.D.S., MS.**
Baltimore, Md., and Farmington, Corm.
Or
Smith
Orthodontic forces can be treated mathematically as vectors. When more than one force is applied to a tooth, the
forces can be combined to determine a single overall resultant. Forces can also be divided into components in
order to determine effects parallel and perpendicular to the occlusal plane, Frankfort horizontal, or the long axis
of the tooth. Forces produce either translation (bodily movement), rotation, or a combination of translation and
rotation, depending upon the relationship of the line of action of the force to the center of resistance of the tooth.
The tendency to rotate is due to the moment of the force, which is equal to force magnitude multiplied by the
perpendicular distance of the line of action to the center of resistance. The only force system that can produce
pure rotation (a moment with no net force) is a couple, which is two equal and opposite, noncolinear but parallel
forces. The movement of a tooth (or a set of teeth) can be described through the use of a center of rotation. The
ratio between the net moment and net force on a tooth (M/F ratio) with reference to the center of resistance
determines the center of rotation. Since most forces are applied at the bracket, it is necessary to compute
equivalent force systems at the center of resistance in order to predict tooth movement. A graph of the M/F ratio
plotted against the center of rotation illustrates the precision required for controlled tooth movement.
Key words: Force, moment, couple, center of resistance, center of rotation, orthodontic tooth movement
T
he literature on orthodontic biomechanics
usually concerns either specific applications of interest
to clinicians or basic questions primarily of interest to
researchers. Few articles have attempted to explain
biomechanical principles by an approach that would
allow the clinician without a background in engineering
to understand the concepts and their potential for clinical relevance. In this article, we attempt to review for
the clinician the basic relationships between forces and
tooth movement.
SCALARSANDVECTORS
Physical properties (such as distance, weight, temperature, and force) are treated mathematically as either
scalars or vectors. Scalars, including temperature and
weight, do not have a direction and are completely
described by their magnitude. Vectors, on the other
hand, have both magnitude and direction. Forces may
be represented by vectors. In addition to magnitude,
direction and point of application must be taken into
account. Direction consists of two properties-a line of
action and a sense.
In order to indicate all of these features, force vectors are represented as arrows (Fig. 1). The length of
*Associate Professor and Director,
Orthodontics,
University of Maryland Dental School.
**Professor and Head, Department of Orthodontics,
University
School of Dental Medicine.
294
Department
of Connecticut
of
the arrow is proportional to the magnitude of the force.
The actual length is arbitrary and will vary with each
particular diagram, as with the scale on a map. In one
case each millimeter of length might be equal to 10
grams and in another case, to 100 grams. What is essential is that all the forces in a given diagram have the
same scale, particularly if they are to be graphically
added or subtracted. Dimensionally, forces are the
product of mass and acceleration and are measured in
units such as newtons or poundals. This complexity is
irrelevant to clinical needs, and the magnitude of forces
can be measured in common units of weight (such as
ounces) or mass (such as grams).
The point of application of a force is indicated by
the origin of the arrow, which in Fig. 1 is at the
bracket. This is simply the point of contact between the
body being moved and the applied force.
Direction is indicated by the body of the arrow
itself and the arrowhead. Without the head of the arrow, the body alone indicates the line of action. The
sense is determined by which end we put the arrowhead on.
Almost every force applied in clinical conditions
will have effects in three planes of space. Throughout
this review, we will deal only with the idealized and
simplified analysis of forces and movements in two
dimensions. Many mechanical principles become much
more complex when three-dimensional aspects are
considered.
Mechanics of tooth movement 295
Volume 85
Number 4
Point of
Application
cl
b
Magnitude
Fig. 1. Forces may be treated as vectors and are convenienUy
represented as arrows. A force vector is characterized by four
features: magnitude, point of application, line of action, and
sense.
Fig. 2. The parallelogram method of determining the resultant
of two forces with a common point of application.
Different
RESULTANTSANDCOMPONENTSOF
ORTHODONTIC
FORCE SYSTEMS
Teeth are often acted upon by more than one force.
Since the movement of a tooth (or any object) is determined by the net effect of all forces on it, it is necessary to combine applied forces to determine a single net
force, or resultant.
At other times there may be a force on a tooth that
we wish to break up into components. For example, a
cervical headgear to maxillary molars will move the
molars in both the occlusal and distal directions. It may
be useful to resolve the headgear force into the components that are parallel and perpendicular to the occlusal
plane, in order to determine the magnitude of force in
each of these directions.
Resultants
from forces
point of application
with a common
In combining forces, the simplest case is for two
forces that have a common point of application (Fig. 2,
A). In order to determine the resultant (combined effect
of the two forces), the two vectors are considered to be
sides of a parallelogram. If the parallelogram is completed (Fig. 2, B), the resultant is the diagonal. Its
length indicates the magnitude of the resultant force on
the same scale as the original forces (Fig. 2, C).
It is important to understand that the resultant will
have the identical effect on movement of the tooth as
the two separate forces. A tooth or a set of teeth moves
in response to the net effect of all forces. If the resultant
is the same, the movement will be the same, regardless
of how many forces are applied or the direction of
individual forces.
For combining more than two forces that have a
common point of application, a series of successive
parallelograms is constructed. Each time, the resultant
from any two forces replaces those forces and is used to
construct the next parallelogram. The sequence in
which forces are combined is of no consequence.
points
of application
Different forces on a tooth are not usually applied
at the same point, as was assumed in the previous
example.
Use of the parallelogram method to combine forces
applied at different points depends upon one of the
basic laws of static mechanics: The law of transmissibility of force. This law can be stated as follows:
When considering the external effects of a force on a
rigid body, the force may be considered to have a
point of application anywhere along its line of action.
The law of transmissibility of force indicates that,
to combine forces, a point of application may be selected anywhere along the line of action of the force.
The line of action may be extended forward or backward off the tooth to construct a totally artificial point
of application in space. The combination of two forces
with different points of application is illustrated in
Fig. 3.
Resolving
a force
into components
Rather than combine two or more forces into a single resultant, it is often useful to divide a single force
into components at right angles to each other. Usually,
the objective is to determine how much force is being
delivered perpendicular and parallel to the occlusal
plane, Frankfort horizontal, or the long axis of the
tooth.
Fig. 4, A illustrates the force on a maxillary canine
from a Class II elastic. How much of the force is acting
to retract the canine, and how much to extrude it?
In this case, the parallelogram procedure for combining forces is reversed. Let us consider the Class II
elastic force vector to be the diagonal of some parallelogram. Because the components are parallel and perpendicular to the occlusal plane (that is, at right angles to
each other), the parallelogram will be a rectangle (Fig.
4, B). When the rectangle is drawn surrounding the
diagonal, the force components are indicated by the
length and width of the rectangle (Fig. 4, C).
Am. J. Orthod.
April 1984
296 Smith and Burstone
Fig. 4. The parallelogram
tical
Fig. 3. The resultant of two forces with different points of application can be determined
by extending
the lines of action to
construct
a common
point of application.
In A, a maxillary
canine is being retracted
with a Class I elastic to a vertical
hook
and a Class II elastic to the bracket.
In B, the lines of action
have been extended
until they intersect.
For the purposes
of
drawing
the parallelogram,
this will be considered
the common
point of application.
In C, the two vectors
are moved along their
lines of action to the constructed
point of application.
The vectors must maintain
their original
lengths
in order to properly
determine
the magnitude
of the resultant.
In D, the parallelogram has been constructed
and the resultant
force obtained.
Its
direction
(both sense and line of action) and magnitude
are now
known.
Since the point of application
is artificial
(and of no
consequence),
the resultant
can either be left as drawn
or
moved forward
on its line of action so that the point of application is on the tooth (E).
With more than one force on a tooth, there are two
methods for determining overall vertical and horizontal
forces. First, the applied forces can be combined into a
single resultant, as described previously, and then this
resultant can be resolved into horizontal and vertical
components. Alternatively, the horizontal and vertical
components for each force can be determined separately, and these components then combined to determine net horizontal and vertical vectors. Direction must
be taken into account. If one force has an intrusive
vertical component and a second force has an extrusive
vertical component, one direction must be selected as
negative and the other as positive, so that addition will
result in the correct net value.
FORCE AND MOVEMENT
Center of resistance
Every object or free body has one point on which it
can (at least in theory) be perfectly balanced. This point
and horizontal
method
components.
for resolving
a force into ver(See text for discussion.)
is known as the center of gravity.* For many physical
calculations, it is as if the rest of the object does not
exist, and all of the weight is concentrated at this single
point.
The movement of a free body depends upon the
relationship of the line of action of the force to the
center of gravity. However, teeth present an additional
complication. They are not free to move in response to
a force; rather, they are restrained by periodontal structures which are not uniform around the tooth (involving
the root but not the crown).
In a restrained body, such as a tooth, a point analogous to the center of gravity is used; this is called the
center of resistance. By definition, a force with a line of
action passing through the center of resistance produces
translation. The center of resistance of a single-rooted
tooth is on the long axis of the tooth, probably between
one third and one half of the root length apical to the
alveolar crest.’ For a multirooted tooth, the center of
resistance is probably between the roots, 1 or 2 mm
apical to the furcation.”
Two important points are evident from the definition of the center of resistance. First, the position of
the center of resistance varies with root length.” Maxillary canines, having longer roots than maxillary lateral
incisors, will have a center of resistance farther from
the bracket. Since (as will be discussed in detail) the
tooth movement resulting from a force delivered at
the bracket depends upon the distance of the line of
action of the force from the center of resistance,
identical forces applied to teeth with different root
lengths can have different effects. A second important
point is that the center of resistance varies with
alveolar bone height. The movement of teeth in
adults with alveolar bone loss will be different than in
*Although technically different, the term ceruer of gravity may be used interchangeably with the term cenfer of mass in all clinical problems.
Mechanics of tooth movement 297
Volume 85
Number 4
Fig. 6. The line of action of the force passes
of resistance.
This tooth will translate,
even
attachment
to the tooth is at the bracket.
Fig. 5. Types of tooth
and C is a combination
Types
of
movement.
A is translation,
of translation
and rotation.
B is rotation,
movement
A tooth can move in one of three general ways: (1)
translation, (2) pure rotation, or (3) combined rotation
and translation. Translation, or bodily movement, occurs when all points on the tooth move an equal distance in the same direction (Fig. 5, A).
Rotation is used here in a restricted sense. As one of
the three basic types of movement, rotation indicates
movement of points of the tooth along the arc of a
circle, with the center of resistance being the center of
the circle (Fig. 5, B).
Any movement that is not pure translation or rotation can be described as a combination of these two
forms of movement (Fig. 5, C).
Effect of forces
Translation: Ifthe line of action of an appliedforce
passes through the center of resistance of a tooth, the
tooth will respond with pure bodily movement (translation) in the direction of the line of action of the applied
force.
Note that the point of application of the force does
not affect its line of action. For example, consider a
vertical hook welded to the bracket of a maxillary
canine (Fig. 6). If the hook has been positioned so that
its height is at the center of resistance, an elastic force
attached to the hook will translate the tooth, even
though the hook itself is attached to the bracket.
Moment of a force. If the line of action of an
through
though
the center
the point of
applied force does not pass through the center of resistance, the force will produce some rotation. The potential for rotation is measured as a moment, and the magnitude of the moment is equal to the magnitude of the
force multiplied by the perpendicular distance of the
line of action of the force to the center of resistance.
Thus, if the force is measured in grams and the distance
in millimeters, the moment of the force will have units
of ‘ ‘gram-millimeters. ’ ’ Fig. 7 illustrates how to determine the perpendicular distance from the line of action to the center of resistance.
Note that either changing the magnitude of the force
or changing the perpendicular distance of the line of
action of the force to the center of resistance has an
equal effect on the magnitude of the moment. Doubling
the force or keeping the force constant but doubling its
distance from the center of resistance will have identical effects on the tendency of the tooth to rotate.
If forces are indicated by straight arrows, moments
can be symbolized by curved arrows. With twodimensional diagrams, clockwise moments will be arbitrarily defined as positive and counterclockwise moments negative. Values can then be added together to
determine the net moment on a tooth relative to a particular point, such as the center of resistance. Point of
application and line of action are not needed; nor are
The direction of a moment can be determined by
continuing the line of action of the force around the
center of resistance, as shown in Fig. 8.
In summary, the moment of a force is equal to the
magnitude of the force multiplied by the perpendicular
distance from the line of action to the point of reference
298
Smith and Burstone
Am. .I. Orthod.
April 1984
Fig. 7. The moment
distance
Fig. 8. The direction
mined by continuing
resistance.
from
of a force is equal to the magnitude
its line of action to the center
of resistance.
of the moment
of a force
the line of action
around
can
the
of the force
multiplied
by the perpendicular
be detercenter
of
and is measured in units such as gram-millimeters. As
with forces, there is only a single net moment on a
tooth, which can be calculated by summing individual
clockwise and counterclockwise moments.
Combined translation and rotation. We now come
to an important and potentially confusing concept. The
forces illustrated in Figs. 7 and 8 will not produce the
“pure rotation” described earlier. Rather, the center of
resistance will rotate but will also translate through
space.* In fact, the translation of the center of resistance will be similar (and perhaps, in theory, identical)
to that which would occur if the force passed through
the center of resistance.
Fig. 9 illustrates two 100 gm forces applied to a
canine. In both Fig. 9, A and Fig. 9, B, there is a distal
force. Both of these forces will have a similar effect in
moving the center of resistance distally. What is different about the forces is that in Fig. 9, B there is also a
moment tending to rotate the tooth, while in Fig. 9, A
there is not. The result is that the type of tooth movement from these forces might be as illustrated. If the
line of action of a force passes through the center of
resistance, the tooth will translate along the line of
action. If the line of action does not pass through the
center of resistance, the center of resistance will translate as if the force did pass through it, but the tooth will
*Technically,
the center of resistance is a point and, by definition, points do not
rotate. The concept of rotation of the center of resistance is a clinically useful
simplification.
Fig. 9. The force in A, passing through the center of resistance,
will result in translation
of the tooth. The force in B, at the
bracket,
will also translate
will cause
a rotation
because
of the moment
created
at the center
of resistance.
also rotate, since the applied force produces a moment
about the center of resistance. The result is a combination of translation and rotation.
Rotation,
couples
Three types of motion were originally defined: pure
translation, pure rotation, and a combination of the
two. Translation occurs when the line of action of the
net force on a tooth passes through the center of resistance, and the combined motion occurs when the line of
action does not pass through the center of resistance.
A single force, therefore, cannot produce pure rotation, meaning a tooth spinning about its center of
resistance. Rather, the only system of forces that can
produce pure rotation of a tooth is called a couple.
A couple consists of two forces of equal magnitude,
with parallel but noncolinear lines of action and opposite senses.
The wire in Fig. 10, A is pictured with a couple.
The forces are the same magnitude, each being 50 gm;
they are parallel to each other but not coincident; and
they face in opposite directions.
In order to demonstrate that this wire will undergo
pure rotation, consider the effects of each force independently and then combine their effects. Each force
will tend to move the center of resistance of the wire
Mechanics of tooth movement
Volume 85
Number 4
Fig. 10. A, Two equal and opposite,
parallel, noncolinear
forces
form a couple.
B, The translational
effects of the forces cancel
each other out, but the moments
of each force combine.
The
result is a moment
with no net force.
,,----.,
I
I//
1
B
\\
Fig. 12. In each figure the solid line indicates
the original position of the tooth. With the center
of rotation
indicated
by the
solid circle,
the type of tooth movement
that could result
is
indicated
by the dotted outlines.
(See text for discussion.)
center
of rotation
t
B
299
\L
I
I
1
C
I
i
I
,/’
Fig. 11. The couple illustrated
is identical
to that in Fig. 10, A
except that it is displaced
20 mm to the right. If the moments
of
the result is identical
to that in
Fig. 10. (See text for details.)
through space just as if its line of action passed through
the center of resistance. The upper force will push the
wire down with a force of 50 gm, while the lower force
will push the wire up with a force of the same magnitude. These translational effects cancel each other
out, resulting in no net translational force.
Will the moments of these forces also cancel each
other out? The magnitude of each moment about the
center will be 50 gm multiplied by the perpendicular
distance of each force to the center of resistance, which
in the example is 10 mm, for a moment of 500 gm-mm.
However, each force produces a 500 gm-mm moment
in the same direction (clockwise), for a total moment of
1000 gm-mm (Fig. 10, B). There is no net translational
force but a large net moment, which will result in pure
rotation. This result also demonstrates that the moment
of a couple is equal to the magnitude of one of the
Fig. 13. A and B represent
the cusp tip before and after movement. A line has been drawn connecting
these points. At the
midpoint
of this line a perpendicular
has been constructed.
The
point at which this perpendicular
intersects
any other perpendicular constructed
in a similar manner
(in the figure the apex
has been selected
as the other point) is the center of rotation.
forces multiplied by the perpendicular distance between
them, or 50 gm x 20 mm in this example.
It will now be demonstrated that it does not matter
where on a rigid object a couple is applied; the external
effect is the same.
Consider a force system of two 50-gm forces separated by a distance of 20 mm, as in Fig. 10, A, except
that, instead of being centered around the center of
resistance, the forces are displaced off to one side
(Fig. 11, A).
300 Smith and Burstone
/-\
Ie?
Am. J. Orthod.
April 1984
center
M
of rotation
vcenter
‘\
of rotation
\
I
(
(
\x
1
\
/
//
/
/4
/ //’
center
of rotation
Fig. 14. The center
of rotation during incisor intrusion
(or extrusion) is perpendicular
to the long axis of the tooth. The amount
of palatal root torque accompanying
the intrusion
decreases
as
the center
of rotation
moves
farther
from the tooth along the
same perpendicular
axis.
As in the symmetrical case, the translational effects
of the vertical forces cancel each other out. The moments of these forces about the center of resistance
equal the magnitude of each force multiplied by its
perpendicular distance to this point. The upward directed force will have a moment of 50 gm x 10 mm, or
500 gm-mm (Fig. 11, B). The downward-directed force
will produce a moment of 50 gm X 30 mm, or 1,500
gm-mm (Fig. 11, C). However, the 500 gm-mm moment is negative (counterclockwise) and the 1,500
gm-mm moment is positive (clockwise). The sum of
these two moments will be 1,000 gm-mm (in a clockwise direction), exactly the same as when each of the
forces was 10 mm away from the center of resistance.
It does not matter where a couple is applied to an
object; the net effect is a moment equal to the magnitude of one of the forces multiplied by the distance
between them. The irrelevance of position is sometimes
stated as the fact that a couple is a “free vector. ”
It is important to distinguish between forces, the
moment of a force, and the moment of a couple. Only
forces are actually applied to an object as a physical
entity. Moments are a measure of the “turning tendency” produced by a force. The moment of a force is
always relative to a point of reference, so that the moment of a force will be low relative to a point close to
I
‘-,
4’
D
Fig. 15. The concept
of a center
define any type of tooth movement
trated in this figure are the centers
first-order
corrections.
of rotation
can be used to
in any plane of space. Illusof rotation
for twc types of
the line of action and high for a point with a large
perpendicular distance to the line of action. A couple is
no more than a particular configuration of forces which
have an inherent moment. This moment is not relative
to any point.
For additional discussions of the concepts reviewed
in this section, we refer readers to the works of Muliigan,4 Hocevar,” and Thurow.6
THE CENTER
OF ROTATION
The movement of a tooth can be described with
more precision than is indicated by the three general
categories of rotation, translation, and combined motion. For a more complete description, the concept of a
center of rotation is used.
If a model of a tooth is attached to a piece of paper
by a pin, the point with the pin in it cannot move, and
this point becomes the center of rotation about which
the tooth can spin. If the pin is placed at the incisal
edge, only movement of the root is possible (Fig. 12,
A); if it is placed at the root apex, movement is limited
to crown tipping (Fig. 12, B). In each case, the center
of rotation is determined by the position of the pin.
Thus, in two dimensional figures, the center of rotation
may be defined as a point about which a body appears
to have rotated, as determined from its initial and final
positions.
The center of rotation can be at any position on or
off a tooth. In Fig. 12, C, for example, when it is a few
millimeters incisal to the apex, some movement of the
root apex will occur in the direction opposite the crown
movement. Retraction of the maxillary canine with insufficient rcot movement has a center of rotation above
the apex of the tooth (Fig. 13). The more nearly translational the movement, the farther apically the center of
rotation would be located. In the extreme case, with
perfect translation, the center of rotation can be defined
as being an infinite distance away.?
Mechanics of tooth movement
Volume 85
Number 4
B
A
Fig. 16. “Stick figures”
can be used
the force and couple are considered
to predict
separately.
C
Centers
of rotation
and tooth
D
tooth movement
and centers
(See text for discussion.)
A simple method for determining a center of rotation is to take any two points on the tooth and connect
the before and after positions of each point with a line.
The intersection of the perpendicular bisectors of these
lines is the center of rotation (Fig. 13).“, 8, s
Although a single center of rotation can be constructed for any starting and ending positions of a tooth,
it does not follow that the single point actually acted as
the center of rotation for the entire movement. The
tooth might have arrived at its final position by following an irregular path, tipping first one way and then
another. As a tooth moves, the forces on it continuously undergo slight changes, so that a changing center
of rotation is the rule rather than the exception. In determining the relationship between a force system and
the center of rotation of the resulting movement, all that
can really be determined is an instantaneous center of
rotation.‘O Nevertheless, this calculation can be clinically useful.
movement
A center of rotation can be determined for any
two-dimensional tooth movement. One of the requirements for thinking mechanically about tooth movement
is to consider the centers of rotation for different types
of tooth movement, in addition to common descriptions
such as bodily movement or torque. Table I shows
some examples of the relationship between typical descriptions of tooth movement and centers of rotation.”
Centers of rotation do not need to be along the long
axis of a tooth. During intrusion, for example, the center of rotation falls on a line which is approximately
perpendicular
to the long axis (Fig. 14). Centers of
rotation can also be defined for rotation of a tooth as
seen from the occlusal aspect (Fig. 15). In an interesting analysis, Hurd and Nikolai” have proposed that a
center of rotation for combined vertical and transverse
movement can be defined as the point which moves the
least on the line of the long axis of the tooth. In agreement with Hocevar,” we prefer not to restrict the center
of rotation to the long axis but, rather, to find the
Table
301
of rotation.
The
effects
of
I
Descriptive
for types
Translation
Uncontrolled
terminology
of movements
(bodily movement)
tipping
Controlled
tipping
Root movement
Position of the
center of rotation
Infinity
Slightly apical to the
center of resistance
Apex
Incisal
(occlusal)
edge
geometric point about which no moment has occurred.
This is more consistent with general use of the term.“, g
However, as NikoW3 points out, the actual movement
of a tooth does involve a series of changing centers of
rotation.
Hocevals suggests that centers or rotation imply
curvilinear motion and that the concept of a center of
rotation has never been proved. However, GoldsteinI
provides proof that such a point exists for any twodimensional movement. Full documentation of twodimensional movement with respect to another body or
a coordinate system requires at least three independent
measurements, since a body moving in a plane has
three degrees of freedom.s One such set of measurements consists of the two coordinates of the center
of rotation plus the angle of rotation about that center.”
Another set, as suggested by Hocevar,” is a translation
vector plus an angle of rotation. A third set involves
translation vectors for two points on the object. All of
these approaches to describing movement are valid and
useful, although a description of motion using the
concept of a center of rotation is probably the most
common throughout the literature on clinical biomechanics.“, g It should be noted that all of these approaches describe only the final change in position of
an object, not the actual path of movement.
Force
systems
and centers
of rotation
Controlling the center of rotation gives precise
control over the type (but not the extent) of tooth
movement.‘” Since all the forces on a tooth can be
Am. J. Orthod.
April 1984
302 Smith and Burstone
l
center
of rotation
e
e
_,_.
....._.,,,
_..
.. .-_,
zb
C
......--._
0
JHHI
B
Fig. 17. The couple is in the opposite direction to the couple
illustrated in Fig. 16. As a result, the apex moves less than the
crown, and the center of rotation is apical to the tooth.
summed to a single force, and all the moments to a
single moment, it follows that a single force plus a
single couple can produce any type of tooth movement-in other words, any center of rotation. In this
section we will review the relationship between the net
moment and the net force at the center of resistance and
the resulting center of rotation.
The extreme situations have been discussed: If the
net effect on the tooth at the center of resistance is a
force only, the center of rotation will be “at infinity”
(the tooth will translate), and if there is a net moment
only, the center of rotation will be at the center of
resistance (the tooth will rotate). The only means of
producing a moment with no net force is with a couple.
For any tooth movement other than these simple
conditions, a force and a couple will be present at the
center of resistance.
In order to understand these combined effects,
“stick diagrams ’ ’ are helpful. Fig. 16, A illustrates a
tooth with a force and couple at the center of resistance.
The general type of movement that will occur can be
estimated by replacing the tooth with a simple line (Fig.
16, B) and applying the force and couple separately. In
Fig. 16, C the effect of the force alone would be to
translate the stick, and in Fig. 16, D rotating the already translated stick in the direction indicated by the
couple will result in more root movement than crown
movement compared with the original position.
Let us consider the effects of changing the ratio
between the force and the couple. Suppose we start
with a pure force producing translation (Fig. 16, C) and
C
Fig. 18. Teeth move according to the forces and movements
acting at the center of resistance. Most orthodontic forces are
applied to the tooth at the bracket. Understanding the relationship between force systems at the bracket and the center of
resistance requires using the rules for equivalent force systems.
The force in A is equivalent to the force system in C, not to the
one in 8.
then add a very small counterclockwise couple. With
the force only, the center of rotation is an “infinite”
distance away. As the couple increases, it brings the
center of rotation in toward the tooth from infinity. If
the couple is large enough relative tc?the force, the
amount of translation could become almost negligible
in comparison to the rotation, and the center of rotation
would be near the center of resistance. With the magnitude of the counterclockwise couple increasing, the
center of rotation moves in from beyond the occlusal
aspect of the tooth.
If the couple added in Fig. 16, C produced a
clockwise moment rather than a counterclockwise one,
the center of rotation would be apical to the tooth rather
than incisal. Again, as the couple increases, the center
of rotation moves closer, but this time, it is approaching from beyond the apex (Fig. 17).
EQUIVALENT
SYSTEMS
The preceding discussion has primarily concerned
forces applied at the center of resistance and couples.
Occasionally, attachments are used that allow forces to
act directly in this area. For the most part, however,
forces are applied on the crown of the tooth. It has
already been mentioned that the moment of a force with
respect to the center of resistance depends upon the
perpendicular distance of its line of action to the center
of resistance, and therefore otherwise identical forces
placed at different positions on the tooth have different
effects on tooth movement. In order to determine how a
tooth will move, it is useful to evaluate the force system
Mechanics of tooth movement 303
Volume 85
Number 4
f-3
1400g-mm
Q
1 ZOOg-mm
.a4
rc
1200g-mm
+
0
ZOOg-mm
=
f-a
1400g-mm
12mm
14mm
1oog
&
OOQ
A
B
Fig. 19. Force systems at the apex (or any other arbitrary point) change when bracket height is altered
(See text for discussion.)
at the bracket to determine the equivalent force system
at the center of resistance.
Consider a distal force at the bracket of a maxillary
canine (Fig. 18, A). This tooth will not exhibit pure
translation, because the force also produces a moment
about the center of resistance. Thus, Fig. 18, B, where
the force has been moved to the center of resistance, is
not equivalent to Fig. 18, A because if a force is applied
as in Fig. 18, B, this tooth will translate. What else
needs to be applied at the center of resistance in Fig.
18, B to have the tooth move the same way that it
would from the force in Fig. 18, A? The answer is a
clockwise couple (Fig. 18, C). The size of this couple,
measured in gram-millimeters, depends upon the magnitude of the force in Fig. 18, A and its perpendicular
distance to the center of resistance. Thus, the force
systems pictured in Fig. 18, A and Fig. 18, C are
equivalent, meaning that they would produce identical
movement. There are three simple rules that allow the
calculation of equivalent force systems. Two force systems are equivalent if (1) the sums of forces in the x
direction are identical, (2) the sums of forces in the y
direction are identical, (3) the sums of moments about
any point are identical.
Before proceeding with some examples, the third
requirement for equivalence requires a brief explanation. So far, the moment of a force has been measured
as the perpendicular distance to a specific point, the
center of resistance. But the moment of a force can be
calculated about any point by simply measuring the
perpendicular distance from the line of action of the
force to the point in question and calculating the moment as the measured distance multiplied by the magnitude of the force. The third requirement listed for
equivalent force systems requires that at any point,
the moments be equal in magnitude and direction.
Some examples should serve to clarify the concept of
equivalence.
8mm
1200g-mm
B
Fig. 20. For bodily movement (translation), a pure force is required at the center of resistance. The equivalent force system
at the bracket requires a large moment from a couple to counteract the moment of the force.
Fig. 19 illustrates two identical maxillary canines.
The canine in Fig. 19, A is being retracted with a
lOO-gm Class I elastic force. The manner of retraction
has been satisfactory, and we would like to continue,
but in rebonding the tooth after the bracket breaks, the
bracket has been placed 2 mm more gingival (Fig. 19,
B). If the fact that there will be a slight change in
angulation of the Class I force is discounted, does the
force system have to be changed in order for the tooth
to continue moving as before? In other words, what is
the force system that will result in equivalent tooth
movement at the new bracket position?
The first rule of equivalence is that the sum of
forces in the x direction is equal in both systems. In
Fig. 19, A there is a lOO-gm distal force in the x direction, so in Fig. 19, B there must be one also. Next, the
forces in the y direction must be equal; there are none in
Fig. 19, A so there must be none in Fig. 19, B. Finally,
the sums of moments about any point must be equal.
304
Am. J. Orrhod.
April 1984
Smith and Burstone
Fig. 21. If the moment of the couple in Fig. 20, B were not large enough to counteract the moment of the
force, there would be a net moment at the center of resistance (Fig. 21, A). The resulting tooth
movement is analyzed in B and C, in which the force produces translation and the net moment
produces rotation. The larger the underestimation of the required couple, the greater the resulting
rotation (D).
1400g-mm
A
B
C
Fig. 22. The previous illustration (Fig. 21) demonstrated the consequences of underestimating the
counterclockwise couple needed for bodily movement. If the couple is overestimated, as in this figure,
the rotation of the tooth is in the opposite direction
Arbitrarily, we can sum moments about the apex. If in
Fig. 19, A the bracket is 14 mm from the root apex, the
moment of the force about the apex will be 100
gm x 14 mm or 1,400 gm-mm. But when a lOO-gm
force is applied to the bracket in Fig. 19, B, the moment of the force about the apex will be 100 gm x 12
mm or 1,200 gm-mm. To make Fig. 19, A and B
equivalent, an additional 200 gm-mm clockwise moment about the apex must be added to Fig. 19, B. A
single force cannot be used to produce this moment
because it would violate one of the first two requirements for equilibrium. With a couple, however, the x
and y resultant forces are unchanged, and the couple
has the same effect wherever it is applied. Thus, a
system equivalent to Fig. 19, A would involve application at the bracket of the same lOO-gm force plus the
200 gm-mm couple (Fig. 19, C).
If the bracket in Fig. 19, B had been placed 2 mm to
the occlusal instead of 2 mm to the gingival, the
lOO-gm force would have a moment of 1,600 gm-mm
about the apex. To make the system equivalent to Fig.
19, A where there is a 1,400 gm-mm clockwise moment, a 200 gm-mm counterclockwise couple would be
required.
We now return from the academic exercise of computing moments about the apex to the clinically important issue of the relationship between force systems at
the center of resistance and force systems at the
bracket. For bodily movement, the force system at the
bracket must be equivalent to a force with no couple at
the center of resistance. Fig. 20, A illustrates the desired force system. What force system applied at the
bracket is equivalent to a pure force at the center of
resistance? First, forces in the x and y directions must
be equal to those in Fig. 20, A, so a 150-gm force at the
bracket is needed in the x direction. The final requirement is that the moments about any point be equal. To
make calculations simple, let us consider moments
about the center of resistance. The force in Fig. 20, A
has no moment about the center of resistance, so that
the system applied at the bracket must do the same. But
the 150-gm force at the bracket, multiplied by its 8 mm
distance to the center of resistance, results in a 1,200
gm-mm moment. This moment of the force is in a
clockwise direction. To make the net moment about the
center of resistance equal to zero, a 1,200 gm-mm
couple in a counterclockwise direction must be added,
as in Fig. 20, B. Thus, the 150-gm force at the bracket
Mechanics of tooth movement
Volume 85
Number 4
305
..-.
...‘....._
c3
A
B
Fig. 23. The force system in A is rarely indicated in edgewise
technique. Almost all centers of rotation for maxillary canine
retraction result from a moment of a couple in the opposite
direction to the moment of the force, as in B.
0
5
10
M/F
plus a 1,200 gm-mm counterclockwise couple will produce pure bodily movement in this example.
but its moment was not quite enough, perhaps 1,000
gm-mm instead of 1,200 gm-mm. At the center of resistance the net effect would be as in Fig. 21, A. The
net 200 gm-mm moment at the center of resistance will
be clockwise, because the 1,000 gm-mm counterclockwise couple does not completely balance the
clockwise 1,200 gm-mm moment from the applied
150-gm force.
Stick figures can be used to explain how this tooth
will move; Fig. 21, B shows the translation due to the
force, and Fig. 21, C shows the rotation due to the
couple. Note that in comparing the before-movement
and after-movement positions, the apices are closer together than the crowns. The center of rotation is apical
to the center of resistance of the tooth. If the moment
had been underestimated by 400 gm-mm rather than
only 200 gm-mm, there would be more of a rotational
effect (Fig. 21, D) and the center of rotation would
move occlusally. Next let us consider the effects of
eliminating the counterclockwise couple altogether, so
that the full 1,200 gm-mm clockwise moment due to
the 150-gm force operates at the center of resistance.
Since the center of resistance “feels” a 150-gm translatory force and a 1,200 gm-mm clockwise moment,
the tooth rotates more than it translates and the center of
rotation approaches the center of resistance.
Now consider what would happen if the counterclockwise couple had been slightly overestimated, so
that 1,400 gm-mm was applied instead of the 1,200
ratio
15
20
25
30
at the bracket
Fig. 24. This graph summarizes the relationship between the
couple-to-force ratio at the bracket and the center of rotation for
a hypothetic tooth with a center of resistance to bracket distance of 10 mm. The directions of the couple and the force are
as in Fig. 23, B. The M/F ratio has units of mm. (See text for
discussion.)
gm-mm needed to counteract the moment of the force
(Fig. 22, A). In this case there is a net counterclockwise
moment at the center of resistance (Fig. 22, B). As Fig.
22, C illustrates, there will be both translation and rotation, but the rotation will produce less crown movement than root movement, so that the center of rotation
will be somewhere to the occlusal of the tooth. As this
couple gets larger, the center of rotation will move in
from the occlusal direction, eventually becoming just
occlusal to the center of resistance.
Because it is the force system at the center of resistance that determines how a tooth moves, and because
the line of action of forces applied at the bracket is
usually a large distance from the center of resistance
(thereby producing a large moment), simple forces
applied at the bracket produce large rotation effects.’
Different centers of rotation along the long axis of a
tooth are created by changing the magnitude of a couple
in the direction opposite the moment of the force about
the center of resistance. For example, in retracting a
maxillary canine, it would be unusual to add a
clockwise couple (Fig. 23, A). As far as the tooth is
concerned, at its center of resistance there is already a
large clockwise moment when a single force is applied
at the bracket. In edgewise technique, all variations of
306
Smith and Burstone
Am. J. Orthod.
April 1984
canine retraction, including pure bodily movement,
more crown than root movement, more root than crown
movement, etc., are achieved by a force and a
counter-tipping couple (Fig. 23, B).
It should be emphasized that the magnitude of
forces and couples cannot be directly compared. The
translation resulting from a 150-gm force has no relation to the rotation resulting from a 150 gm-mm couple.
In fact, roughly 300 to 400 gm is an effective force
level for mesial translation of a mandibular molar, but a
couple with a moment of about 2,000 gm-mm is necessary for efficient molar uprighting.
MOMENT-TO-FORCE
TOOTH MOVEMENT
RATIOS
AND
From the preceding discussion, it follows that the
type of movement exhibited by a tooth is determined by
the ratio between the magnitude of the couple (M) and
the force (F) applied at the bracket.‘. ‘IL ” In terms of
direction, the moment of the couple is almost always
going to be in the direction opposite the moment of the
force about the center of resistance.
Note that moments are measured in gram-millimeters and forces in grams, so that a ratio of the two has
units of millimeters (this represents the distance away
from the bracket that a single force will produce the
same effect). Unfortunately, it has become conventional in orthodontics to ignore these units and just
speak of the moment-to-force ratio as a pure number.
It is possible to graph the relationship between the
center of rotation and the M/F ratio applied at the
bracket. This is for the situation in which a horizontal
force (perpendicular to the long axis) is applied at the
bracket along with a couple producing a moment in the
direction opposite the moment of the force. Adapted
from Burstone and Pryputniewicz,’ Fig. 24 illustrates
these relationships for a tooth with a lo-mm distance
from the bracket to the center or resistance. The
abscissa (x axis) gives the moment-to-force ratio at the
bracket, and ordinate (y axis) gives the distance from
the center of resistance to the center of rotation (in this
example, the units of the M/F ratio are millimeters).
When only a force is applied at the bracket (M/F
ratio of zero), the center of rotation is just apical to the
center of resistance (labeled A on the graph). The resulting tooth movement is uncontrolled tipping. As the
moment of the couple increases, the center of rotation
moves apically (toward the position labeled B). With a
distance from the bracket to the center of resistance of
10 mm, the center of rotation approaches infinity as the
M/F ratio approaches 10/l (position C). As soon as the
M/F ratio passes 10/l, the net moment at the center of
resistance changes direction, since the moment of the
couple is now greater than the moment from the applied
force. The center of rotation is incisal to the center of
resistance. At first it is an “infinite” distance to the
incisal (position D). When the M/F ratio is 12 or 13 to
1, the center of rotation will be at the incisal edge
(position E, root movement), and as the M/F ratio increases up to about 2011, the center of rotation becomes
just incisal to the center of resistance (position F), approaching a purely rotational movement.
This graph has important implications.” It is the
ratio between the applied couple and force that determines the type of tooth movement, not the absolute
magnitudes.“’ (This mechunical principle does not take
into account the fact that magnitudes of forces and
couples are important in determining the biologic response to an orthodontic force system.) Except for M/F
ratios which result in centers of rotation near the center
of resistance, small changes in the M/F ratio have important consequences on the type of tooth movement.
An M/F ratio of about 8/ 1 will put the center of rotation
at the apex, tooth movement being controlled tipping,
while an M/F ratio of 1211 will put the center of rotation at the incisal edge, which is usually termed root
movement. Contrast these two M/F ratios with an M/F
ratio of 10/l, which will result in bodily movement.
Obviously, small changes in this ratio have major effects on clinical tooth movement.
SUMMARY
Undesired or inefficient tooth movement during
orthodontic treatment results from individual variations
in biologic response and the improper use of forces.
Application of the rules of biomechanics allows one of
these sources of variation to be reduced or eliminated.
The ability to measure and control couple-to-force
ratios at the bracket is a key to predictable and controlled tooth movement.
We thank David Gipe for preparation of the illustrations
and Barbara Bass for typing several drafts of the manuscript.
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RJ: Holographic
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Reprint
requests to:
Dr. Richard J. Smith
Department
of Orthodontics
University
of Maryland
Dental
666 W. Baltimore
St.
Baltimore,
MD 21201
School
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