moment force ratios and controlling root are presented

1. Moment-to-Force Ratios;
Prof Dr Maher Fouda
Mansoura
Egypt
applied to orthodontics

2. Moment-to-Force Ratios and
Controlling Root
Position
The second and commonly
employed method of controlling
tooth movement with fixed
appliances is with alteration
of moment-to-force ratios, which
is a description
of the relationship between the
applied force to move
a tooth and the counterbalancing
force couple (moment)
required to prevent the rotational
tendency produced as a
result of the applied force being
at a distance from the centre
of resistance.
Center or resistance
of a single root tooth,
Center of resistance
of a molar
Any force acting through the center of
resistance of a tooth will
make it translate in a bodily manner.

3. Moment-to-Force Ratios and
Controlling Root
Position
What does this mean?
The force applied to a
tooth at the
bracket will lead to a
moment round the
centre of resistance
of the tooth; this is
called the moment of
the force, which
will be in the direction
of the applied force.
(A) The moment of a force is equal to the magnitude of the
force multiplied by the perpendicular distance from its line of
action to the center of resistance. (B) The direction of the
moment of a force can be determined by continuing the line
of action around the center of resistance.

4. Moment
In physics, a moment refers
to the turning effect
produced
by a force acting at a
distance on an object,
usually defined
with respect to a fixed
reference point .
In mechanics, the moment
of a force describes the
tendency of a force applied
to a body to produce
rotation.
A force acting at a
distance from the
centre of mass
of a free body in a
vacuum will lead to
rotation about the
centre
of mass, as well as a
small amount of
bodily translation in
the
direction of the
applied force.

5. The application of
orthodontic forces is typically
at the crown of a tooth, and
thereby the applied force
is not through the tooth’s
centre of resistance. A force
applied at a distance from the
centre of resistance will
not produce bodily movement
only, but will also result
in rotation. In orthodontic
biomechanics, the moment,
or more accurately the
moment of force, describes
the
tendency of a force applied to
a tooth to produce rotation.
The moment of a force (MF) is equal to
the product
of the force magnitude and the
perpendicular distance (Ma) from
its line of action to the centre of
resistance (red circle). The point
of force application in both (a) and (b)
is the maxillary canine
hook, but, for comparative reasons,
the forces have different
lines of action. The perpendicular
distance from the line of
action of each force to the centre of
resistance is known as the
moment arm (Ma).

6. The magnitude of this rotational
effect produced by a
force acting at a distance is
expressed as the product of the
force and the perpendicular
distance from the line of action
of that force to the centre of
resistance. The direction of
the moment may be found by
following the line of action
round the centre of resistance
towards the point of force
application .
The direction of the moment
may be found by
following the line of action
round the centre of resistance
towards the point of force
application.

7. The perpendicular
distance from
the line of action of
the force from the
point of force
application
to the centre of
resistance is termed
the moment
arm (or lever arm) .
The perpendicular
distance from the
line of action
of the force from
the point of force
application (here
the point of
force application is
the bracket slot) to
the centre of
resistance is
termed the moment
arm (Ma), also
sometimes referred
to as the
lever arm.

8. In classical mechanics,
the unit is the newton
metre (N⋅m) or newton
centimetre
(N⋅cm), but in
orthodontics the gram
force-millimetre
(gf-mm) unit is more
commonly employed (1
gf-mm =
0.98N⋅cm).

9. The magnitude of the moment of a force may be
determined
by two variables:
• the magnitude of the
force
• the distance from the
point of force application
to the
centre of resistance.
The line of action of any force (F)
not passing through the center of
resistance creates a moment (M),
which is a rotational or tipping
effect on the tooth. According to the
formula M = F × d, a moment is
proportional to the magnitude of
force and the distance (d)
perpendicular from its line of action
to the center of resistance

10. It is within the clinician’s
capability to manipulate
either
of these variables in order
to achieve the desired
tooth
movements.
The edgewise orthodontic
system can simultaneously
deliver forces, moments
and counterbalancing
moments.
Forces applied at a distance from the
centre of resistance result in rotation of
the tooth. This should be considered in all
planes of space: (i) buccolingual, (ii)
mesiodistal and (iii) occlusal.

11. Couple
In mechanics, a
couple refers to a
pair of equal and
parallel
forces acting in
opposite directions
(i.e. opposite lines of
action), and tending
to cause rotation
about an axis
perpendicular
to the plane
containing them.
An archwire in a bracket slot can be used to
apply a couple in the: (i) mesiodistal, (ii)
buccopalatal, and (iii) occlusal planes. (iv) The
couple created between the bracket slot and
archwire can be used to control the tipping
caused by the moment of the force (F). In this
way, bodily tooth movement can be achieved

12. A couple is a moment where
the sum of the forces is zero.
The magnitude of a
couple is the product of the
magnitude of one of the
forces
multiplied by the
perpendicular distance to the
opposite
force . In order to determine
the direction of
the rotation, either force may
be followed round the centre
of resistance and towards
the origin of the opposite
force.
The magnitude of a force couple is
the product of
the magnitude of one of the forces
multiplied by the
perpendicular distance to the
opposite force. For example, the
diagram shows the occlusal view
of a premolar tooth which is to
be rotated using a force couple. If
the forces shown are each
60 g (a buccal elastic force applied
to the bracket and an equal
and opposite palatal force applied
to a bonded button), and the
distance (d) is 9 mm, the
magnitude of the force couple
(moment
of the couple, MC) would be 60 × 9
= 540 gf-mm.

13. The moment created
by a couple is always
round the
centre of resistance,
no matter where the
pair of forces is
applied . As the
distance between the
two
forces of the couple
decreases, the overall
magnitude of
the couple also
decreases.
(a) The moment created by a
force couple is always
round the centre of resistance,
no matter where the pair of
forces
is applied. A pair of equal and
opposite forces is shown acting
at
a distance on a beam, resulting
in rotation round its centre of
resistance. (b) As the distance
between the forces decreases,
the
magnitude of the force couple
also decreases, but rotation still
occurs round the centre of
resistance. It should be noted
that the
pair of forces applied to the
beam in this diagram resembles
the
position of a torquing force
couple applied to a bracket on
the
crown of a maxillary incisor.

14. The clinical
implication for fixed
orthodontic
appliances is that
no matter where a
bracket is
positioned on the
crown of a tooth,
the moment of the
couple
will lead to rotation
of the tooth round
its centre of
resistance.
c) The clinical implication for
fixed
orthodontic appliances is that
no matter where a bracket is
positioned on the crown of a
tooth, the moment of the force
couple will lead to rotation of
the tooth round its centre of
resistance. In this diagram, a
rectangular archwire with
palatal
root torque applied is engaged
into the bracket slot on a
maxillary central incisor. The
torsional stress of the engaged
archwire places a force couple
in the bracket slot. The moment
of this force couple (MC) will
result in rotation of the maxillary
incisor round its centre of
resistance and thereby a change
in the
tooth’s inclination.

15. The magnitude of the couple
depends on the magnitude
of the forces and the
distance between the two
forces, with
the moment of the created
couple being the sum of the
moments created by each of
the two forces. Where the
two forces creating the
couple act on effectively
opposing
sides of the centre of
resistance, their effect is
additive.
A couple created by two equal and opposite forces
acting on a tooth. The total moment (MC) is the vector
addition of the two moments (m1, m2) generated by
the two forces (F1, F2). Here, m1 = F1 × d1, m2 = F2 ×
d2. Because the two moments are in the opposite
direction, one of the moments will be assigned a
negative sign and the other positive. The net moment
(M) will be obtained by adding the two: M = m1+ (−m2)

16. This applies to round or
rectangular orthodontic
archwires
changing the angulation
of a tooth by engaging in
a bracket,
and to rectangular
archwires changing
the inclination of a
tooth by engaging in a
bracket .

17. The magnitude of the couple
depends on the
magnitude of the forces (shown
as two equal and opposite red
arrows acting on the bracket) and
the distance between the two
forces, with the moment of the
created force couple being the
sum of the moments created by
each of the two forces. Where
the two forces creating the couple
act on effectively opposing
sides of the centre of resistance,
their effect is additive. This
applies to the settings illustrated
in (a) and (b).

18. (a) Round or
rectangular orthodontic archwires
change the angulation of a
tooth by engaging in a bracket slot. The
expression of
mesiodistal tip, i.e. the change in the
angulation of the tooth,
stops when the archwire becomes
passive in the bracket slot. (b)
Rectangular archwires change the
inclination of a tooth by
engaging in a bracket slot. The
expression of buccolingual
torque, i.e. the change in the inclination
of the tooth, stops when
the archwire becomes passive in the
bracket slot.

19. To counteract
the effect of the
moment of the
applied force, a
force couple
may be generated
at the bracket,
creating a new
moment,
called the moment
of the couple,
which can
counterbalance
themoment of the
force .

20. If the moment of
the
couple is opposite
the moment of the
force, and equal in
magnitude,
rotational
movement will be
prevented,
permitting
bodily movement.

21. Moment-to-force ratios. The force applied to a tooth
(F) will lead to a moment round the centre of resistance (CR) of
the tooth; this is called the moment of the force (MF), which will
be in the direction of the applied force. To counteract the effect
of the moment of the applied force, a force couple may be
generated at the bracket by engagement of an archwire, creating
a new moment, called the moment of the couple (MC), which can
counterbalance the moment of the force. If the moment of the
couple is opposite the moment of the force, and equal in
magnitude, rotational movement will be prevented, permitting
bodily movement.

22. The following examples describe the biomechanics. (a) Application of a
single point force
(F) to the crown of a maxillary central incisor. This force may be
from a labial spring or from a palatally directed elastic; in either
case, there is a palatally directed retroclining force on the
crown. The moment of the force (MF) will result in rotation
round the centre of rotation, which in this situation is almost
identical to the CR of the tooth, with the crown moving in the
direction of the applied force.

23. (b) If a force couple is applied at
the bracket on the crown of the incisor tooth by engaging a
rectangular archwire, the tooth will rotate round its CR, leading
to a change in the inclination of the tooth.

24. (c) However, if a
palatal force is applied, as in (a),
and a counterbalancing force
couple is applied at the bracket,
as in (b), the combination of the
applied palatal force (and its
moment, MF) and the force
couple
at the bracket (and its moment,
MC) will mean that rotation of
the tooth does not occur round
its CR (red circle). The position
of
the centre of rotation (blue
circle) will alter based on the
ratio of
the MF in relation to the MC .

25. Relationship between moment-to-force ratio and the
type of tooth movement.
The ratio between the moment of the
applied force (MF) and the counterbalancing moment of the couple (MC)
determines the type of tooth movement that will occur. This
important concept requires further explanation in relation to a common
orthodontic situation. Consider the maxillary arch in the
space closure stage of fixed appliance treatment. A continuous archwire
is in place, and a space-closing force is applied to move the
maxillary incisors backwards towards the posterior anchor teeth. This
force F is acting at the brackets. The force F will lead to a
moment round the centre of resistance; this moment of the force is
termed MF. In the following four examples, the magnitude of the
force F and the moment of this force, MF, will remain unchanged. (a) If the
archwire is a relatively thin round wire, there will be no
couple created at the bracket, i.e. MC = 0. The centre of rotation (Crot, blue
circle) will be very close to and just apical to the centre of
resistance (CR, red circle), and the force F will lead to rotation round Crot.
The incisors retrocline by simple tipping, i.e. the crown
moves in the direction of the force and the root rotates in the opposite
direction.

26. Relationship between moment-to-force ratio and the
type of tooth movement.
(b) A rectangular archwire is placed in the bracket.
The space-closing force F remains the same, and MF thereby remains
the same. The rectangular wire in the rectangular slot will lead
to the formation of a force couple in the bracket, which is the moment
of the couple (MC). If the bracket prescription is with palatal
root torque, or if the archwire has palatal root torsional stress applied
to it, MC will be the counterbalancing force to MF. As long as
MC is less than MF, the tooth crown will still rotate in the direction of
the original force F, but because of MC, Crot will move apically.
There is less root movement and more crown movement, which is
termed controlled tipping. The incisors are still retroclining, but the
crowns are moving a greater distance in a palatal direction, with
minimal root movement. If Crot moves to the apex, the apex will not
move, and the crown will retrocline the greatest distance.

27. Relationship between moment-to-force ratio and the
type of tooth movement.
(c) If a larger rectangular archwire is placed, or
greater palatal torsional root
‘torquing’ forces are applied, such that MC
increases until MC = MF, the tendency to
rotation is eliminated; i.e. because MC is the
counterbalancing force to MF, the two moments
cancel each other, and there is no rotation
round CR. Crot moves apically towards
infinity, and the effect of the original force F is
to bodily translate the incisors in the direction
of the force F, i.e. bodily retraction.

28. Relationship between moment-to-force ratio and the
type of tooth movement.
(d) If
even greater torsional forces are placed
by the rectangular archwire in the
bracket slot, such that MC becomes
greater than MF, Crot
moves incisally, and there will be greater
root movement in a palatal direction. If
Crot moves to the incisal tip, the tip will
not move,
and the root will ‘torque’ the greatest
distance in a palatal direction.

29. The ratio of the moment of the couple to the original
applied force will determine the type of tooth movement
that occurs; this is the moment-to-force ratio .
Varying the moment-to-force ratio (by changing the magnitude
of the applied force and/or the force couple at the
bracket) allows the location of the centre of rotation of
a tooth to be altered along its long axis, thereby giving
control over the type of tooth movement .

30. Relationship between moment-to-force ratio and
the type of tooth movement. The ratio between
the moment of the
applied force (MF) and the counterbalancing
moment of the couple (MC) determines the type
of tooth movement that will occur. This
important concept requires further explanation
in relation to a common orthodontic situation.
Consider the maxillary arch in the
space closure stage of fixed appliance
treatment. A continuous archwire is in place,
and a space-closing force is applied to move the
maxillary incisors backwards towards the
posterior anchor teeth. This force F is acting at
the brackets. The force F will lead to a
moment round the centre of resistance; this
moment of the force is termed MF. In the
following four examples, the magnitude of the
force F and the moment of this force, MF, will
remain unchanged.

31. (a) If the archwire is a relatively
thin round wire, there will be
no
couple created at the bracket,
i.e. MC = 0. The centre of
rotation (Crot, blue circle) will
be very close to and just apical
to the centre of
resistance (CR, red circle), and
F will lead to rotation round
Crot. The incisors retrocline by
simple tipping, i.e. the crown
moves in the direction of the
force and the root rotates in
the opposite direction.

32. (b) A rectangular archwire is placed in the bracket.
The space-closing force F remains the same, and
MF thereby remains the same. The rectangular
wire in the rectangular slot will lead
to the formation of a force couple in the bracket,
which is the moment of the couple (MC). If the
bracket prescription is with palatal
root torque, or if the archwire has palatal root
torsional stress applied to it, MC will be the
counterbalancing force to MF. As long as
MC is less than MF, the tooth crown will still rotate
in the direction of the original force F, but because
of MC, Crot will move apically.
There is less root movement and more crown
movement, which is termed controlled tipping.
The incisors are still retroclining, but the
crowns are moving a greater distance in a palatal
direction, with minimal root movement. If Crot
moves to the apex, the apex will not
move, and the crown will retrocline the greatest
distance.

33. (c) If a larger rectangular archwire is
placed, or greater palatal torsional root
‘torquing’ forces are applied, such that
MC increases until MC = MF, the
tendency to rotation is eliminated; i.e.
because MC is the
counterbalancing force to MF, the two
moments cancel each other, and there is
no rotation round CR. Crot moves
apically towards
infinity, and the effect of the original
force F is to bodily translate the incisors
in the direction of the force F, i.e. bodily
retraction.

34. (d) If
even greater torsional forces
are placed by the rectangular
archwire in the bracket slot,
such that MC becomes greater
than MF, Crot
moves incisally, and there will
be greater root movement in a
palatal direction. If Crot
moves to the incisal tip, the
tip will not move,
and the root will ‘torque’ the
greatest distance in a palatal
direction.

35. There is a direct relationship
between the magnitude
of the applied force and the
magnitude of the
counterbalancing
couple, in that the heavier the
applied force to the
crown of a tooth, the larger the
moment of the force, and
thereby the larger the moment
of the counterbalancing
couple within the bracket
required to prevent tipping.
A couple created by two equal and opposite forces
acting on a tooth. Te total moment (MC) is the vector
addition of the two moments (m1, m2) generated by
the two forces (F1, F2). Here, m1 = F1 × d1, m2 = F2 ×
d2. Because the two moments are in the opposite
direction, one of the moments will be assigned a
negative sign and the other a positive sign. Te net
moment (M) will be obtained by adding the two: M =
m1 + (–m2).

36. The orthodontic
biomechanical mechanism for
achieving
such a system is a fixed
bracket or other attachment
on
the tooth crown, constructed
such that forces may be
applied
at two points on the tooth.
This concept was elucidated
by
Calvin Case in 1921 based on
round wires .
A, Te moment created by
a couple is always around
the CRES or CG (MC = F ×
D). B, No matter where
the pair of forces is
applied, the couple
created will always act
around the CRES or CG. As
the distance between the
two forces decreases (d <
D), the overall magnitude
of the couple decreases
(mc < MC).

37. The concept of bodily movement from a
force couple applied to the crown was
described by Calvin Case in 1921. (a)
Here,
Case is describing a force being applied
at point i, accomplished by attaching to
the crown a rigid ‘root-wise extension or
bar’, in order
for the line of force to be sufficiently
above the ‘point of greatest resistance’
at c. He suggests that a ‘more or less
bodily movement’
would occur in the direction of the force,
but that this would not be with the
absolute certainty that would follow the
more ‘scientific
control of the force for this character of
movement, described later’.

38. (b) The image marked A represents Edward Angle’s method. The
force couple is represented by the two arrows at the bracket. The upper
arrow represents the ‘centre of work’ or ‘fulcrum’ (i.e. centre of
resistance). The images marked B–D are Case’s gradual modifications
of the appliance in order to decrease ‘the distance to the area of
work or alveolar resistance, both of which greatly increase the
mechanical advantage’.

39. (c)
Combination
s used to
create
couples for
bodily tooth
movement.

40. Alternatively, an auxiliary spring
may be used together
with a round base archwire. The
auxiliary spring would be
required to place a force on the
facial surface of the incisor
crown gingival to the bracket,
with the force in a palatal
direction, causing it to rotate
round its transverse axis
(which is the usual reason for
using such auxiliary springs,
i.e. palatal root ‘torquing’
forces).

41. However, in this situation,
this force would need to be opposing the
retroclining
force at the bracket on the crown, the
result being bodily
retraction. Such torquing springs are used
with traditional
Begg appliances, but can be formed for
use with some
edgewise systems, and are rarely
employed for the type
of movement described here (Figure 2.31)
(see Chapter
11, Figures 11.23 and 11.24), as the use of
rectangular
archwires has made such complex
mechanics unnecessary.
(a, b) Palatal root ‘torquing’
auxiliaries on the
maxillary incisors, being used with a
traditional Begg appliance.
Source: courtesy of Dr David Spary.

42. The most common technique
and mechanism for the
application of a force couple
is to use a rectangular
archwire
ligated into the rectangular
edgewise bracket slot
in order to generate the
moment required to control
the
incisor inclination during
retraction. The two points of
contact are the opposite
edges of the rectangular
archwire
within the bracket slot .
A couple is
applied to
derotate a
premolar.
Before (A) and
after (B)
Generation of a
force couple by the
interaction between
the bracket slot and
the archwire

43. The magnitude of the couple
depends on the
magnitude of the forces (shown as
two equal and opposite red
arrows acting on the bracket) and
the distance between the two
forces, with the moment of the
created force couple being the
sum of the moments created by
each of the two forces. Where
the two forces creating the couple
act on effectively opposing
sides of the centre of resistance,
their effect is additive. This
applies to the settings illustrated in
(a) and (b).

44. (a) Round or
rectangular orthodontic archwires
change the angulation of a
tooth by engaging in a bracket slot.
The expression of
mesiodistal tip, i.e. the change in
the angulation of the tooth,
stops when the archwire becomes
passive in the bracket slot. (b)
Rectangular archwires change the
inclination of a tooth by
engaging in a bracket slot. The
expression of buccolingual
torque, i.e. the change in the
inclination of the tooth, stops when
the archwire becomes passive in
the bracket slot.