Taken from the JCO 1979 Oct (676-683): Common Sense Mechanics: Part 2 -

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Common Sense Mechanics 2

THOMAS F. MULLIGAN, DDS

Forces and Moments

We all know what a force is, but

sometimes we tend to confuse the

relationship between force and

moment. Both are extremely

important to us as they produce the

movements we seek, as well as those

we consider undesirable. Simply

stated, a force is nothing more than a

"push" or "pull," and acts in a straight

line (Fig. 15). Whenever this line of

force passes through the center of a

body— in orthodontics we refer to the

Center of Resistance— there is no

moment produced and therefore no

rotational tendency (Fig. 16A). When a

force acts away from the center, a

moment is produced and a rotational

tendency occurs (Fig. 16B).

A moment is the product of force

times distance. If the line of force does

not pass through the center of

resistance of the tooth, then there is a

distance between this line of force and

the center. It is the perpendicular

distance from this line of force to the

center that causes the moment on the

tooth, resulting in rotational

tendencies (Fig. 17). Although I don't

care to use numbers or specific

magnitudes in tooth movement, the

magnitude of the moment is

determined by this force times the

perpendicular distance to the center.

We could double the force and cut the

distance in half, or double the distance

and cut the force in half, and in both

cases we would produce the same

moment or rotational tendency (Fig.

18).

What does all of this mean? First of all,

the orthodontist cannot think of forces

and torques (moments) as being the

same. You can "sense" a force when

you bend a wire, but you cannot

"sense" torque. Because the latter is

simply a product of force times

distance, as previously discussed, the

distance (length) is just as effective as

the force. If the force passes through

the center of resistance, no

perpendicular distance is involved.

Therefore, regardless of the

magnitude of the force, there is no

moment (Fig. 19). Force times zero

distance always equals zero. We might

use a lot of force and produce no

moment or a small moment, while a

small force might produce a large

moment due to the distance involved

(Fig. 20). So, beginning right now, it is

important to get used to treating the

two as separate entities. One is a

product of the other. Because all of

this is going to become essential, later,

when we discuss differential torque

for anchorage and non-anchorage

problems, and because ultimately you

will see that the clinical application is

simple, fast, and easy to understand, it

is critical that the groundwork be laid

beforehand.

I am going to go through a step-by-

step discussion of what I have, over

the years, referred to as the "Cue Ball

Concept." If we can think of things in a

way that relates to some of our

personal experiences in life, I think you

will find it much easier to understand

and ultimately apply. After all, if it

cannot be applied, then this is strictly

academic and a waste of your valuable

time.

Cue Ball Concept

Anyone who has had the experience of

playing pool has held a cue stick and

applied a force on the cue ball.

Experience taught the individual

where to strike the cue ball in order to

produce a given response. If we

desired English, we applied a force off

center (Fig. 21). We produced left or

right English at will, simply by deciding

to apply the force to either the left or

right side of center on the cue ball.

If we only wished to "translate" the

cue ball— move it in a straight line

with no left or right English— we

applied the force right through the

middle of the cue ball (Fig. 22). By the

way, with a tooth we use the term

Center of Resistance, whereas, in a

free body we use the term Center of

Mass. Obviously the ball rotated or

rolled forward due to the friction of

the table, but the response was

predictable. A force applied through

the center resulted in straight line

movement with no left or right English

(moment). Since we know from

experience, therefore, how to predict

a response based on the point of force

application, let us take a step-by-step

look at the reasons behind this

predictability.

Translation

Again, if we apply a force through the

center of the cue ball, it will move

forward in a straight line (Fig. 23).

Unlike the tooth, this is a free body

with a set of rules we will discuss.

Whenever a force passes through the

center of such a body, the body will

translate. There will be no rotation—

other than the forward roll due to the

friction of the table itself. The reason

there is no rotation (moment) is that

the line of force has no perpendicular

distance to the center; the force is

passing through the center. So, we can

make the statement that a force acting

through-the center of such a body

produces translation without rotation.

This is a predictable response based on

a known point of force application.

Rotation and Translation

If we take exactly the same force and

apply it on the same body, but instead

of applying it through the center,

apply it off center, then we create a

situation where the line of force has a

perpendicular distance from the

"Center of Mass" (a free body

expression). This means that we now

produce not only translation, but also

rotation, as a result of the moment

produced (Fig. 21). As we know from

experience, this is exactly what

happens when we decide to strike a

cue ball to the left or right of center. A

force applied on a body, but not

through the center of that body,

results in translation and rotation.

Pure Rotation (Couple)

Although when we play pool, we do

not apply two forces on a cue ball at

the same time, we could do it to prove

a point. If we were to apply two forces

on the cue ball, equal and opposite, in

the same plane of space, the ball

would not translate in any direction.

Instead, it would simply maintain its

position and "spin" (rotate) (Fig. 24).

The reason for this is that the two

forces cancel each other out, but leave

a net moment (rotation) due to the

fact that each of these "Lines of Force"

acts at a perpendicular distance from

the center of the ball. Now that we

have "played" the game of pool

together, I hope we can see the

reasoning behind our experiences and

from this learn that it is possible to

predict a response based on a known

point of force application.

Forces and Moments Acting on Teeth

With some of these basics behind us,

let us take a look at tooth movement

when we attempt to accomplish more

than one type of tooth movement at a

time. In the previous material, it was

shown how the force and direction can

be determined by whether the bend is

in the center or off center. Therefore,

if we use a tipback bend for overbite

correction, as is done in a number of

techniques today, we can certainly

recognize that when the short

segments are placed into the molar

tubes, the long segments, prior to

bracket engagement, lie in the muco-

labial fold (Fig. 25A). From this we can

see that the long segment points

apically in the incisor area and

therefore indicates an incisor intrusive

force while the molars have an

extrusive force present.

But, there is more to it than just these

forces. What about the moments?

When the wire is brought down from

the mucolabial fold for insertion into

the incisor brackets (Fig. 25B), the

force required acts at a perpendicular

distance from the center of resistance

in the molar (Fig. 25C), thus producing

mesial root torque or distal crown

thrust on each of the molars involved.

When the wire is engaged into the

incisor brackets, the intrusive force

acts in a straight line and usually

passes labial to the center of

resistance in the incisors (Fig. 26). This

produces a smaller moment that on

the molar, because in spite of the fact

the forces are equal, the distances

involved are radically different.

So, when the archwire is tied into

place and tied back at the molar tubes,

we have significantly different

(relatively) magnitudes of torque (Fig.

27) which we can refer to as

"differential torque". If we do not tie

the archwire to the molar tubes, and if

friction does not accomplish the same

by causing binding at the tubes, the

anterior and posterior moments may

be permitted to respond

independently of each other. If tied

back, the system behaves as a whole,

and the "tug of war" is apparent with

the molar having the obvious

mechanical advantage with the larger

moment. The clinical applications of

differential torque will be discussed

later .

Thus far, we see a force system as

illustrated in Figure 28. But, again, that

is not all that is taking place. Let us

take a look at a distal view of the

molar teeth and keep the cue ball

concept in mind (Fig. 29). If the wire is

round, instead of rectangular, and

permitted to "roll" inside the tubes,

the extrusive force present on the

molar teeth then acts at the molar

tubes which lie, usually, buccally to the

center of resistance in these teeth.

This force times distance results in

molar lingual crown torque. So we can

begin to see that such torque is not

necessarily dependent on the use of

rectangular wire. Torque is simply a

product of force times distance and

does not recognize the type of wire

involved. Incidentally, if a wire were

very rigidly attached to the tubes, the

applied force would pass lingual to the

center of resistance, thereby inducing

buccal crown torque instead .

When one observes an effect, he

should be able to interpret the cause

and vice versa. We should also begin

to recognize that such force systems

should not be routinely considered as

undesirable side effects except for the

orthodontist who is unaware of their

presence and therefore is not

prepared to prevent undesirable

effects as well as to utilize the systems

effectively when indicated. If lingual

crown torque is desired, it should be

permitted to act. If undesirable, it can

be prevented with a lingual arch, a

rectangular wire, or whatever means

the operator chooses.

So, looking at the force system, thus

far, we recognize molar extrusive

forces, incisor intrusive forces, molar

mesial root torque significantly

(relatively) larger than the incisor

lingual root torque, and molar lingual

crown torque. Does this seem

complicated? It might for anyone used

to concentrating only on the single

force or moment desired, but the

entire system exists, whether we like it

or not— not just the portion with

which we are concerned. In any case,

awareness of the entire system will

afford us many exciting opportunities

as we will see later. We will discover

that there are means available for

utilizing parts of the system while

overcoming other parts, because we

will be dealing with such matters as

forces of occlusion, cusp heights, wire

size and lengths, etc., whereby we can

learn to control force magnitudes so

that although an extrusive component

of force might be present on a molar

and considered to be undesirable, it

can be prevented from acting and

therefore not become a threat. Force

systems will always be present, but

not all phases will be permitted to

respond.

Lingual Root Torque

Now, after all of the previous

discussion which involved a tipback

bend, we are able to become

reasonably familiar with the force

system involved. Let us take a look at

other bends in the same archwire that

begin to affect the force system. If we

place lingual root torque into the

incisor section, we produce a long

segment and a short segment (Fig. 30),

just as was the case with the tipback

bend. The long segment indicates a

molar intrusive force and therefore an

extrusive force on the incisors. We can

also see that the torque produced on

the incisors is a result of force times

distance, since the long segment has

to be brought down to the molar tube,

and the force required to bring it down

acts at a perpendicular distance to the

incisors (Fig. 31). If the long segments

from the tipback bends maintain the

same angular relationship as the long

segments from the incisor torque

bend, the vertical forces cancel each

other and only moments remain.

Therefore, no overbite correction may

occur even though we might expect it.

The anterior lingual root torque

introduces a vertical component of

force that must be considered .

If the long segments just discussed are

unequal in angular relationship, then

the one producing the greater angle

relative to the level of the archwire

will determine the net force present.

For example, if lingual root torque

produces the greater angle as shown

in Figure 32, the net forces will be

intrusive on the molar and extrusive

on the incisor. Therefore, if we are

hoping for overbite correction, but

increased our lingual root torque to

this point, we can expect our overbite

to increase instead of decreasing. So,

we might decide, if we know this

beforehand, to either increase the

molar tipback bend, decrease the

amount of lingual root torque on the

incisor segment, or a combination of

each, in order to assure ourselves of a

net intrusive force on the incisor

segment for overbite correction.

Recognition of the problems and

intelligent decision making will only

follow a thorough understanding of

the underlying principles.

(TO BE CONTINUED)

Fig. 26 The

intrusive force

acting through

the incisor

bracket usually

lies labial to the

center of

resistance, thus

producing a

moment, but

smaller than

the one on the

molar.