Common Sense Mechanics 76pgs. MUlligan

Introduction 1

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INTRODUCTION l REMARKS: Many things will be said in this book that you may never have heard before – not even in your graduate education. Having been interested in cause-effect relationships since my childhood, it has always been my nature to constantly ask the question, “WHY?” Often, I did not receive answers that satisfied my curiosity, so from the time I began my practice in August 1962, I began searching for the answers myself. Having a deep interest in the clinical aspects of orthodontic treatment, I closely observed both expected and unexpected tooth movements. I enjoy the academics in orthodontics, but find it disappointing when the academic and clinical aspects are not in agreement. I sincerely hope that everyone who reads this book will find both their personal and professional lives enhanced. When illustrations are shown, some will be used only as visual titles and/or enhancements to the subject and will not be labeled as Figure 1-1, Figure 1-2, etc . However, YOU MUST illustrations that apply to treatment will REMEMBER THIS receive such labels. Therefore, I will begin this entire discussion on “Common Sense Mechanics” by initiating the very Don’t memorize, understand. basic first level. Do not consider it offensive if seemingly basic subjects are Understand!!!

Introduction 1

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included. When completely understood, it will be seen what biomechanics can really do for the individual who is interested in clinical applications that are effective as well as exciting.

Cognitive Dissonance The inability to believe what you have been programmed not to believe, however compelling the evidence

I don’t want to challenge what you believe. What I do want to challenge – and perhaps change – are some of the things you do not believe.

Cognitive Dissonance

Changing Your Beliefs

Looking at cognitive dissonance above, the definition and statement that follow should clarify the intent behind the material published in this book. When teeth move in directions which are unexpected,

Let’s take a look at the effects of archwire resiliency. there must be a reason

WHY. Why Do Teeth Often Move in Unexpected Directions? Frequently, teeth are expected to move in one direction, but often move in completely unexpected directions. Unfortunately, this often leads to the use of rigid appliances to prevent such

Introduction 1

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undesirable responses. Transpalatal and lingual arches may find themselves being used routinely in order to avoid undesirable consequences. It will later be shown that such appliances can most often be avoided. In fact, in over 45 years of practice, the author has never used either. Let’s now take a look at some of the so-called visual problems that are encountered in mechanics. These visual interpretations are quite common in the profession and are largely responsible for the failure to predict and recognize correct force systems. The first examples will deal with the characteristic known as archwire resiliency.

Can you visually determine what moment will become present at the cuspid bracket?

Figure 1-1

This is the Correct Moment

Figure 1-3

Incorrect Moment on the Cuspid

Figure 1-2

The reason lies in archwire resilience.

Figure 1-4

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In observing Figures 1-1 thru 1-5, it can be seem that the actual moment on the cuspid creates a force system which can be very beneficial for many types of treatment, such as anchorage control and Class II correction. Note that teeth in Response the buccal segments have been by-passed. The greater the interFigure 1-5 bracket distance, the greater will be the resiliency created by any given archwire. Now, let’s take a look at the occlusal view of archwire bends. Whether resiliency occurs in the sagittal or occlusal plane makes no difference. The confusion usually remains the same.

Changing planes of space should not alter the prediction of force systems. What forces (disregard the moments) do you predict in the occlusal plane of space?

Planes of Space

Figure 1-6

Note in Figures 1-6 thru 1-8 that the archwire has been activated by bending the wire lingual to the molars. Determine only the forces present and disregard the moments for these examples.

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Figure 1-7 Figure 1-8 The wire in each case is bent in the same direction with forces that are completely different. It will be seen later that the location of the bend is significant. Now, let’s look at a somewhat different situation, but still involving resiliency and its effects. In this case, consider only the moments and not the forces for Figure 1-9.

What moments will be present at the molar tubes? (Disregard the forces).

Are these moments correct?

Figure 1-9 Figure 1-10 It would appear that the resulting moments in Figure 10 are correct because of the arc through which the wire must travel for engagement with the molar tubes. Wires are normally placed into the molar tubes first, thereby eliminating the visual effect of resiliency.

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The moments shown in Figure 1-11 are correct. These are simple examples of the effects of archwire resiliency. More dramatic effects will be seen later when the Wire/Bracket relationships are discussed in detail.

Are these moments correct?

Figure 1-11 Next, a discussion will take place involving the problems associated with visual inspection in the sagittal plane of space. Determine only the forces present at this time. Same wire as below

Same wire as below

Same wire as above

Same wire as above

Figure 1-12

Same as below

Same as above

Figure 1-13

Figure 1-14

Each wire is bent in the same direction.

Only forces shown

Moments not shown

Figure 1-15

In Figures 1-12 thru 1-15, it can be seen that although the wires are bent in the same direction, it is the different location of each of the bends responsible for the entirely different forces produced on the molars and the cuspids.

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Next, the same illustrations will be shown, but only the moments are to be determined. Disregard any forces present.

Figure 1-16

Figure 1-17

Only Moments shown

Figure 1-18

Forces not shown

Figure 1-19

Again, look at the completely different directions and magnitudes of the moments involved in Figures 1-16 thru 1-19. With forces and moments having been determined individually, the entire force systems can now be seen in Figure 1-20. This is not easy for many individuals to predict.

Total Force Systems

Forces and Moments

Figure 1-20

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Remember also!

Remember this! Regardless of whether the smaller moment is clockwise, counterclockwise, or nonexistent – the net moment determines the forces present.

The long section points in the direction of the force produced. Therefore, the short section points opposite to the force produced. The bracket located closest to the bend contains the largest moment.

Figure 1-21

Figure 1-22

Figures 1-21 and 1-22 offer very simple rules to predict the forces and moments previously discussed. This is not exactly a scientific method for identification, but it is a good memory method until such time as immediate recognition of forces and moments takes place. This will occur with a thorough understanding of equilibrium requirements. With an understanding of the above rules, it can be seen in Figures 1-23 thru 1-29 that all the bends shown are really offcenter bends. Later, when the center bend is discussed, it will be seen that it can be considered to be a variation of the offcenter bend. These similarities will be extremely helpful in predicting force systems.

Molar Tip-Back

Cuspid Root Torque

Figure 1-23

Molar Tip-Back

Lingual Root Torque

Figure 1-24

Molar Tip-Back

Labial Root Torque

Figure 1-25

Introduction 1

Molar Tip-Back

Toe-In Bend

Figure 1-26

Molar Tip-Back

In-Bend

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Molar Tip-Back

Toe-Out Bend

Figure 1-27

Molar Tip-Back

Out-Bend

Figure 1-28 Figure 1-29 What has been discussed thus far has been an elementary introduction to force systems. The purpose is to illustrate the unreliability of using visual inspection. Contrary to what is taught, P.E. is not as effective as O.P.E. (Figures 1-30 & 1-31).

PERSONAL EXPERIENCE

P.E.VS.O.P.E. Figure 1-30

VS. OTHER PEOPLE’S EXPERIENCE

Fig 1-31

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THE SHORT STORY Orthodontic treatment involves the use of many appliances used for control. Transpalatal and lingual arches are in common use today because of unexpected tooth movements. With the use of such additional appliances, rigidity interferes with the need for teeth to be placed into the neutral zone, as rigidity overcomes the natural forces produced by the muscles and function. An approach has been presented to aid the orthodontist in recognizing why these forces seem somewhat difficult to recognize. Archwire resilience and archwire shape can be very misleading in the prediction of force systems. Because the typical orthodontist has been taught to read force systems by visualizing the relationship of the archwire to the bracket slot prior to insertion, incorrect force systems are often anticipated. No force system is produced until an archwire is placed securely into the bracket, at which time characteristics such as resilience come into play. Resilience is an important factor to recognize as it will increase or decrease with changing interbracket distances. Wire shape has also been shown to create force systems other than what might be anticipated. Explanations and solutions to the above problems have been presented at a relatively elementary level at this point. With the material to follow, the orthodontist will be able to predict force systems in a more sophisticated manner. What will be presented may be far different than what you have learned in the past, but it will offer new opportunities for the clinician.

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SUGGESTED READINGS Mulligan TF. Common sense mechanics. Phoenix: CSM, 1040 East Osborn Road, 1982. Smith RJ, Burstone CJ. Mechanics of tooth movement. Am J Orthod 1984;85:294-307.

Understanding Forces and Moments 2

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UNDERSTANDING FORCES AND MOMENTS 2 For some reason, the terms forces and moments do 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 which follows 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 FORCE SYSTEMS personal experience that a specific force system does not necessarily produce the same response for different patients. Nothing in life The same force system may produce a variable response. happens without a reason. Force Force magnitude is a significant factor. 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.

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Teeth may rotate Incisors may intrude

Molars may erupt.

Forces may produce variable response.

Figure 2-2

Forces may produce variable response.

Forces may produce variable response.

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 a straight line. Forces consist of a push or pull.

Figure 2-5 Figure 2-6 In Figure 2-5, a force is applied through the center of mass, a term 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 describe rather than define a force, it can be seen in Figure 2-6 that a force acts 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 we push from 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 it passes 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.

Forces act in a straight line – not a curve

1

2

3

Push from the lingual

Pull from the buccal Figure 2-7

Understanding Forces and Moments 2

MOMENTS (FxD) Moments are produced as a result of forces acting away from the Center of Resistance or Center of Mass.

Figure 2-8

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ROTATION The product of force x distance produces the moment on the body. Therefore, 1/2 the force x twice the distance produces the same moment as 1/2 the distance x twice the force.

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 distance established 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 at twice 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 of pool 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, let’s 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 ball in question is known as the cueball 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.

For those who are unaware, the cue-ball is the “white ball.”

Visualize the crown of a tooth as a cue-ball

The Cue Stick represents the force that will be applied to the brackets and tubes of the teeth. Cue Stick

This represents the “Point of Force Application”

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 application shown 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 right English. Naturally, the ball will roll down the table due to friction, but disregard this rotation.

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CUE-BALL CONCEPT

1.

A force applied through the center of a body will cause the body to move in a straight line and in the same direction as the applied force.

Figure 2-13

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.

2. A force applied away from the center of a body will cause the body to move in same previous direction, but rotation will also occur as a result of the moment created by the line of force acting at a perpendicular distance to the center of the body.

Figure 2-14 There are three possible movements that may occur, just as in the real world of orthodontics. 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 2-14 where 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. Regardless of the where the equal and opposite forces are applied, the body will undergo pure rotation around the center of resistance. Let’s see where this concept applies at the clinical level.

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Translation The result of the applied force is a moment and a force at the center of resistance. Center bend producing equal & opposite moments to those already present.

This is an Equivalent Force System.

Figure 2-16 In Figure 2-16 upper left, forces have been applied at the crown level 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 result is that only pure forces remain as seen on the right in Figure 216. 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 217. 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

Understanding Forces and Moments 2

Tipping Moments Eliminating the Tipping Moments

Figure 2-17

A closing force at the brackets produces tipping moments eliminated by Center Bends.

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degrees. All adjustments for increasing or decreasing moments for the proper force system are created by adjusting whatever closing mechanism is in use, such as coil springs or powerchain. In Figure 2-18, tipping the incisors together would not be acceptable. The placement of a center bend into the wire produces 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 or powerchain 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 cueball concept regarding translation and rotation.

Understanding Forces and Moments 2

Figure 2-19

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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 relationships are 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 been created. 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 force produced 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 minimize chairside 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. Many other 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 a couple - 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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Following surgical exposure seen in Figure 2-23, elastics were utilized to create equal and opposite forces (couple) once the cuspid was brought into alignment (Figure 2-24). The lingual bracket had been placed at the time of exposure as no other surface of the tooth was available for bonding. Simply adding a labial bracket later afforded the opportunity to provide a couple.

Couple Required

PURE ROTATION

Applied Couple

Remember to visualize the crown as a Cue-Ball

Figure 2-25 Figure 2-26 Although there is no difficulty in treating the above rotation with another approach, Figure 2-25 demonstrates the application of a couple in providing the correction seen in Figure 2-26. Figure 2-27 will provide the final example for pure rotation. Following space closure, center bends have been placed to provide for equal and opposite moments at each bracket in order to parallel the roots. Figure 2-27 While discussing forces and moments, we should look at the effect of vertical forces acting through the molar tubes. Undesirable consequences often occur as a result. Figure 2-28

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shows that intrusive forces may cause buccal displacement of molars due to buccal crown moments produced. Intrusive force produces buccal crown moment.

Extrusive force produces lingual crown moment.

Figure 2-28

IMPORTANT! As the upper molars are widened, the curve of Monson increases and no longer harmonizes with the curve of Wilson.

Intrusive force produces buccal crown moment.

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 play a role. When such undesirable movements do occur, an easy solution is provided by the utilization of molar control bends to 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 is this 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.

Understanding Forces and Moments 2

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Functional Curves Spee

These curves can be helpful in determining which arch is involved and to what degree.

Monson

Wilson

The long axis should lie parallel to the Internal Pterygoid resulting in axial loading (stability).

2 Curve of Monson

1

3 Curve of Wilson

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 shown that 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 have not 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.

Understanding Forces and Moments 2

Lingual Crown Displacement

1

Buccal Crown Displacement

Original Molar Width

2

3

Change in Molar Width

2nd Molar Width is Normal

Figure 2-32

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1

Original Molar Width

2

3

Change in Molar Width

2nd Molar Width is Normal

Figure 2-33

In Figure 2-32 #1, the eruptive Second Molars are in force has caused the first molars to move lingually as normal position. observed in #2. In #3, the second molars have now erupted. It remains important to know which of the molars Second Molars are in normal transverse dimension. are out of position. In Figure 2-33 the same series of events Figure 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

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curves may result in functional problems involving axial loading and leading to later instability. This concludes the chapter regarding forces and moments. What may 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. In addition 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-arch solutions for many malocclusions that will help to avoid placing appliances on opposing 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 produce variable 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. Different types 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.

Static Equilibrium and its Importance 3

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STATIC EQUILIBRIUM AND ITS IMPORTANCE 3 A

B

1

Figure 3-1

Everyone has had the experience in life of being in balance or out-ofbalance. Whether balance has been lost due to excessive drinking or playing on a teeter-totter while young, the experience of imbalance at one time or another is certainly universal in nature.

The following few illustrations will A demonstrate balance and imbalance. B No one will question the outcome of two individuals seated on the teetertotter as shown in Figure 3-1. If two 2 individuals are of equal weight and equal distance from the fulcrum in Figure 3-2 Figure 3- 1, they will be in complete balance - a state known as static equilibrium. No one will question this because of personal experience. When we experience an event, we accept it as a fact. This can lead to an interest in learning the cause/effect relationships involved and provide later solutions in orthodontic biomechanics.

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In Figure 3-2, if the same two individuals are of unequal weight and A sitting at the same distance from the fulcrum, balance will only take place following displacement at which time 3 static equilibrium will again be established. No one is going to Figure 3-3 question this outcome because it has undoubtedly been experienced in childhood. B

In Figure 3-3, it is shown that the imbalance that occurred in Figure 3-2 can be restored to a state of balance if the two individuals simply shift their weight so that the larger individual is closer to the fulcrum than the smaller individual. Of course, the opposite is also true. The smaller individual can relocate a greater distance from the fulcrum. No one will question what has been demonstrated in these three illustrations as all are based on personal experience. The wonderful thing about personal experience is that the outcome cannot be questioned. Instead, the cause can be determined with an analysis. The requirements for static equilibrium take place in orthodontic treatment just as they do with the teeter-totter. Because there are no exceptions to static equilibrium when archwires are fully engaged, the orthodontist has the opportunity to determine what forces and moments are present regardless of any visual perceptions that may be very misleading and totally incorrect. Many intelligent orthodontists have come to incorrect conclusions regarding the force systems proposed for specific methods of treatment. A thorough understanding of static equilibrium will change many such conclusions.

Static Equilibrium and its Importance 3

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Let’s begin by seeing why static equilibrium is established in the examples of the teeter-totter. It is important to recognize that there are three requirements for static equilibrium. REQUIREMENTS FOR EQUILIBRIUM 1. Sum of the Vertical Forces equals zero. 2. Sum of the Horizontal Forces equals zero.

Static Equilibrium Figure 3-4

3. Sum of the Moments around a Common Point equals zero.

Figure 3-5

Figure 3-4 beautifully illustrates static equilibrium whose three requirements are stated in Figure 3-5. The first requirement states that the sum of the vertical forces must equal zero. The second requirement states that the sum of the horizontal forces must also equal zero. Finally, the third requirement states the sum of the moments measured around a common point must also equal zero. None of these requirements may be absent in any case regarding static equilibrium.

Figure 3-6

Figure 3-7

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With prescription brackets so widely in use today, forces and moments are produced that must always meet these three requirements. How many orthodontists actually recognize the total force systems produced with various wire/bracket angles as shown in Figures 3-6 and 3-7? Since the design of prescription brackets is to replace the need to bend archwires in such a manner as to produce the desired shape for tooth movement, the forces and moments produced will be those that meet the requirements for static equilibrium. In other words, a specific force system must meet these requirements. Orthodontists often do not recognize the total force systems required for equilibrium and concentrate instead on those forces and moments desired for the particular type of tooth movement in question. In the Class II division 2 malocclusion, forces for overbite correction or moments for torque may be the prime considerations without recognizing that balancing forces will occur that can create as much undesirable response as the tooth movement which has been intended.

Figure 3-8

Figure 3-9

Figure 3-10

The wire/bracket angles shown in Figures 3-8 thru 3-10 will be discussed in great deal when the subject of Wire/Bracket Relationships is discussed. Looking closely at these illustrations, it will be noted that the wire/bracket angles are identical in both upper and lower portions of each figure. In the upper portion,

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the bracket slots are level with bends placed in the archwire, whereas in the lower part of each illustration, the archwires have no bends, but instead the brackets are angulated. Such angles may be purposely introduced into the appliance or may be produced by the malocclusion. In either case, a force system will be produced in order to meet the complete requirements for static equilibrium. If there is any difficulty in identifying the total force system in each of the above, then the unrecognized aspect of the force system may be the cause of undesirable responses that lead to the use of preventive measures such as lingual or transpalatal arches. Figure 3-11 tends to create controversy with those visually oriented in determining force systems, as it appears the smaller moment should Figure 3-11 be clockwise. Such is simply not the case. Contrary, however, to the visual assumption, resilience causes just the opposite. Although resilience has already been introduced as one of the factors that sometimes creates difficulty in recognizing the correct force system, a wire/bracket analysis will now be done to prove what the total force systems must be in order to comply with the three requirements for static equilibrium. Remember again there are no exceptions. A Wire/Bracket Analysis

B

EQUILIBRIUM REQUIREMENTS HAVE BEEN FULFILLED

C x

A

D


Sum

of the vertical forces equals zero. of the horizontal forces equals zero.
Sum of the moments measured around a common point equals zero.
Sum

“If you don’t believe it, you won’t achieve it.”

Figure 3-12

ACTIVATIONAL FORCES

Figure 3-13

A

+

B

+

C

+ D

=

Figure 3-14

0

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In Figure 3-12, it is true that “If you don’t believe it you won’t achieve it.” This analysis should eliminate any doubts regarding the forces and moments present. Only Figures 3-8 and 3-10 will be analyzed as they represent the extremes in the angles created. Beginning with Figure 3-13, forces necessary to engage the wire into each slot are shown. These are referred to as activational forces. It is obvious that two activational forces will be required at each bracket for insertion into each slot. At this point it will be assumed that such forces are all equal in magnitude. This is only an assumption and remains to be proven as all three equilibrium requirements must be met and must be proven. If the assumed forces are all equal, then their sum equals zero and the first requirement for equilibrium has been fulfilled. There are no horizontal forces, so therefore the second requirement has also been fulfilled. Finally, the sum of the moments must equal zero when measured around a common point. Any point can be utilized, but for the sake of convenience a point marked X will represent the point to be used. If each force is now multiplied by the perpendicular distance to this point, it will be seen that their sum is equal to zero. Forces A & D produce equal and opposite moments and the same is true with forces B & C. Therefore, the three requirements have been fulfilled as shown in Figure 3-14. The next step is to complete the activational force system. In Figure 3-15, the force system is determined by taking the already proven equal and opposite forces at each bracket and recognizing that these couples result in pure moments.

Static Equilibrium and its Importance 3

Activational Force System

De-Activational Force System

Figure 3-15

Figure 3-16

35

The center bend or two off-center bends – each producing the same wire/bracket angles – result in the same force systems.

Figure 3-17

The deactivational force system in Figure 3-16 is simply a reversal of the activational force system. Finally, in Figure 317, it is shown that a bend in the center is exactly the same as two off-center bends whenever the angles are equal and opposite. This is a significant relationship which will be utilized in effective clinical treatment at a later point. B

D x

A

C

EQUILIBRIUM REQUIREMENTS ARE NOT FULFILLED
Sum

of the vertical forces equals zero.


Sum

of the horizontal forces equals zero.


Sum of the

moments measured around a common point does not equal zero.

ACTIVATIONAL FORCES

Figure 3-18

A

+

B

+

C

+

D

=

Figure 3-19

THIS ACTIVATIONAL FORCE SYSTEM IS INCORRECT

Figure 3-20

Figure 3-10, shown earlier, will now be analyzed. Looking at Figure 3-18, an assumption has again been made that the activational forces are equal and opposite. Thus, the first two requirements for equilibrium again are met. When summing the moments around a common point as was done before, it can be seen in Figure 3-19 that the third requirement has not been met, as Figure 3-20 shows a net clockwise moment. Therefore, our original assumption of equal and opposite forces cannot be correct. This force system is completely unbalanced, as a couple is required on the system in order to provide a

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balancing moment in the opposite direction of the net moment shown. When the first system was analyzed and the center bend found to produce equal and opposite moments, one moment balanced the other and therefore no forces were required. In this case it can be seen that the net system consists of a clockwise moment. It will be seen with a continued analysis that such balancing forces will be proven to exist. In Figure 3-21, it will now be assumed the forces acting at each bracket are unequal. We are seeking proof that equilibrium requirements are met. B

D X

A

C

EQUILIBRIUM REQUIREMENTS HAVE BEEN FULFILLED


Sum of the vertical forces equals zero.




Sum of the horizontal forces equals zero.




Sum of the moments measured around a

common point equals zero.

ACTIVATIONAL FORCES

Figure 3-21

A

+

B

+ CD +

D A

=

Figure 3-22

0

ACTIVATIONAL FORCE SYSTEM

Figure 3-23

Although the forces are unequal, when all the forces on the system are added, the total forces in the vertical plane of space again equal zero, so the first requirement is fulfilled. Likewise, the second requirement is again fulfilled because there are no horizontal forces present. The third requirement is now met as each force, multiplied by the distance to the common point X, when added now equals zero as shown in Figure 3-22. Therefore, the correct activational force system in Figure 3-23 has been proven DE-ACTIVATIONAL FORCE SYSTEM and now is reversed as shown in Figure 3-24 to show the deactivational force Figure 3-24 system. The deactivational force system is of interest to the orthodontist as it represents tooth movement.

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Now that a wire/bracket analysis has been demonstrated to prove force systems that meet the three requirements of equilibrium, a discussion can take place regarding the force acting at each bracket whenever a net moment is present on the system. Such forces are equal and opposite and constitute a couple. Why must such forces be present? Remember that equal and opposite forces - known as a couple - produce a pure moment. Therefore, the presence of a couple in any force system producing net moments at the brackets simply results in a balancing moment. The balancing moment is opposite in direction to the net moment produced at the brackets. In the following clinical examples, the couple that is required in order to produce balancing moments will be shown. In Figure 3-25, there are counterclockwise The balancing forces moments shown for the worsen the occlusal plane. correction of an apical base discrepancy which result in Figure 3-25 a worsening of the midline. Because this would result in an unbalanced system. a couple is automatically introduced as it provides a pure moment in the opposite direction which is equal and opposite to the moments placed by the orthodontist. The canted occlusal plane worsens. If looking only at the individual force acting at each of the incisor brackets, it can be seen that the two forces result in a clockwise moment. No measuring is involved as this balancing moment is equal and opposite to net moments intentionally

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placed at the brackets by the orthodontist. Equilibrium is something to be respected, as the failure to recognize the complete force system can result in unexpected consequences that may result in an increase in treatment time for the patient not to mention frustration for the orthodontist. Not only will further balancing forces be demonstrated, but it will also be shown that different types of tooth movement may actually require the same force systems – not different force systems as one might expect. In Figure 3-26, two tooth-moving systems are illustrated. A partial appliance is shown with a tip-back bend which on activation will produce an intrusive force for overbite correction while another partial Eg. Rectangular appliance is shown with a or square wire. wire containing twist for Forces & moments lingual root torque when remain the same. Only the activated. nomenclature changes.

Although the objectives Lingual Root Torque is simply a differ, it can be seen that Tip-back Bend “turned around.” one system is simply the reverse of the other. When Figure 3-26 one or the other is simply “turned around,” the force systems are identical. In other words, lingual root torque is simply a tip-back bend turned around. If at all confusing, remember the early rule regarding long and short sections which indicate the direction of the forces present, while stating that the bracket or tube closest to the bend contains the largest moment. This is an effective memory

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system until the subject of mechanics is discussed in more depth. However, understanding is more important than memory. Occlusal planes may be altered by couples present to balance net moments while occlusal planes already in balance with equal and opposite moments require no balancing forces (couples). Altered occlusal planes can be desirable or undesirable. This is a choice that can be made by the orthodontist. Changes may be made to alter the amount of anterior teeth to be shown or made for a number of other reasons including the relationship to the condylar path for those concerned with posterior disclusion of teeth in protrusive movements.

Distal Crown Torque

Lingual Root Torque

Equal & Opposite Moments

Figure 3-27

Figure 3-28

Figure 3-29

Figures 3-27 & 3-28 demonstrate a need for a balancing couple. Figure 3-29 shows balancing moments present and therefore no need for a couple. Couples are created only for net moments. As this chapter comes to a close, another look at Figure 3-1 and Figure 3-3 shown at the introduction to the subject of static equilibrium might now make more sense as to why balance exists when individual weights are equal or unequal. It was first acknowledged that experience provided acceptance of the

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outcome. Now the cause of the outcome can be understood and applied in clinical orthodontics as well. As can be seen, the illustration to the left in Figure 3-30 B shows two individuals with A equal weights seated at equal distances from the fulcrum. 3 When analyzing the force system, it can be determined that the forces and moments that are shown meet the three Figure 3-30 requirements for equilibrium. So, it is not only known from experience that such balance occurs, but now it can be understood why such balance takes place. It is the understanding which is important as it leads to application in other areas – in this case, recognition of important force systems in clinical orthodontics. A

B

1

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THE SHORT STORY Static equilibrium is an often misunderstand subject for the clinician. Such equilibrium has always been part of our lives. Examples experienced in childhood, such as two individuals of equal or unequal weight on a teeter-totter seated in various positions, demonstrates that the results are not in question, but only the cause as it relates to the equilibrium requirements. These same requirements apply to orthodontic treatment during wire/bracket engagement. The orthodontist is interested in what forces and moments are produced as a result of wire/bracket engagement. In static equilibrium, three requirements must be met in order to engage archwires into the bracket slots. These initial forces of insertion are referred to as activational forces and always meet the three requirements of equilibrium. First, the sum of the vertical forces must equal zero. Secondly, the sum of the horizontal forces must equal zero. Finally, the sum of the moments measured around a common point must also equal zero. When only two out of three of these requirements are met, static equilibrium does not exist. When recognizing that the force system meets all three requirements, the system is then in equilibrium. Once the three requirements are fulfilled, the net activational system is determined and then reversed to determine the deactivational system which is of concern to the orthodontist as this is the tooth-moving aspect of the system. It is not uncommon for the orthodontist to arrive at incorrect deactivational force systems and therefore experience unexpected tooth movements which often lead to the use of transpalatal or lingual arches.

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SELECTED READINGS Demangel C. Equilibrium situations in bend force systems. Am J Orthod Dentofac Orthop 1990;98:333-339. Nikolai RJ. Bioengineering analysis of orthodontic mechanics. Philadelphia: Lea & Febiger, 1985:56-69. Mulligan TF. Common sense mechanics. 3. Static equilibrium. J Clin Orthod 1979;13:762-766.

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FORCES AND RELATIVE MAGNITUDES 4 This is an interesting subject for a profession that spends its time with forces and moments on a regular basis. Surprisingly enough, it will be shown that several methods of determining force systems today are simply incorrect. Forces are often measured that are not only inaccurate in magnitude, but also incorrect in the direction in which they act. To make matters even worse, sometimes the forces measured disappear altogether when the archwire is inserted. Any error in the correct determination of force magnitude is automatically an error in the magnitude of any moments produced, as Force X Distance = Moment. No wonder orthodontists wonder why certain movements seem to go nicely at times while at other times such responses seems to diminish or even become nonexistent. It has earlier been mentioned that force systems may vary in response because of differences in force magnitudes. It was illustrated in Chapter I that molars might erupt for one patient, but not for another because of force magnitudes overcoming - or not overcoming - the forces of occlusion. Keeping in mind that nothing in this world happens without a cause, it is important to recognize that when such effects do occur, the causes may be either unknown or misunderstood. It is the purpose of this discussion to help provide an understanding of the factors contributing to the problem.

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44

One of the factors requiring discussion is the cantilever, as this is often the method used in measuring force magnitudes for a patient, thinking that such force magnitudes will exist. First of all, a good example of a cantilever is the diving board shown in Figure 4-1. A cantilever is considered to be a simple beam in the engineering world and is characterized by a pure force at one end and a single moment at Figure 4-1 the point of attachment.The forces of course are equal and opposite at each end.

This is a cantilever measurement.

Figure 4-2 Figure 4-3 A very common method of measuring forces is to take an instrument such as the Richmond gauge and measure an archwire for intrusive forces as shown in Figures 4-2 and 4-3. This presents a problem as the cantilever only exists prior to inserting the wire into the brackets. Following insertion into the brackets the measurement taken is no longer valid, as moments are introduced because of archwire resilience. This is only one of many different factors responsible for faulty measurements.

Determining Forces and Relative Magnitudes 4

Round Resilience creates angles.

As a result, this is not a cantilever.

The force magnitude is no longer that which was measured.

45

In Figure 4-4, it is obvious that the archwire crosses the incisor brackets at an angle, thus creating the presence of moments at these brackets. A cantilever is characterized by only a pure force. Moments cannot be present, as they will alter the force measured..

Figure 4-4 If a true cantilever effect is Cantilever intrusion of cuspids is desired, there are other ways more effective than engaging of accomplishing such an archwire into bracket slots. objective. Figures 4-5 and 4-6 show a practical approach for the use of a cantilever. Although the archwire is inserted into the four incisor (placed under bracket wing) brackets, the cuspid slot is not engaged and the wire is Figure 4-5 placed under the bracket wing as shown. When wire/bracket angles are discussed later, a cantilever system will be introduced by locating a bend at a particular position between the brackets. Figure 4-6 Other illustrations can be seen in Figures 4-7 thru 4-9 which show the opportunity to provide cantilevers in various situations.

Determining Forces and Relative Magnitudes 4

46

Cuspid intrusion required Cantilever – (Incisors excluded)

Cantilever – (Incisors bracketed)

Intruding six teeth - Cantilevered Cuspids

Figure 4-7

Figure 4-8

Figure 4-9

In Figure 4-7, when only cuspid intrusion is required, a continuous archwire can be placed under the cuspid bracket wings as shown here and in the previous illustrations. Note that the incisors have not been bonded at this point. In Figure 4-8, the incisors have been bonded, but the archwire previously shown is stepped gingivally to the four incisors in order to avoid movement of the anterior teeth while avoiding the introduction of a moment at each of the cuspid brackets. Finally, in Figure 49, the archwire has been placed under the cuspid wings and into the incisor brackets which are now level. It can be seen that there are a number of ways in which to provide a cantilever force when desired. With cantilevers, pure forces may be applied to the intended teeth without the presence of moments. It will be seen throughout the various chapters, that the vast majority of orthodontic treatment is being achieved with partial appliances. By the time the various aspects of biomechanics have been fully discussed, it will be seen that partial appliances offer many distinct advantages over full appliances. This is not to say that full appliances (Figure 4-10) cannot be utilized near the end of treatment, but rather to point out that many effective tooth movements can be achieved in the earlier stages of

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47

treatment which would not be achieved if a full appliance were introduced at the initiation of orthodontic treatment.

Avoiding Full Appliances

Taking advantage of partial appliances

Figure 4-10 Figure 4-11 Why are partial appliances (Figure 4-11) then, not in common use? There are several reasons. First of all, very little has been understood in orthodontics regarding the force systems that can be developed with partial appliances. Interbracket distances are significantly increased and resilience increased as a result. This resiliency may introduce moments that can be extremely favorable for anchorage, protraction, and other problems as well. There are those orthodontists who prefer indirect bonding, and would like to place all brackets simultaneously rather than go through a procedure that only allows placing a few at a time. The orthodontic companies are in the business of selling brackets. There are over 1100 prescription brackets on the market and more are being developed. Many different types of brackets are being developed and sold, not only by the companies, but by orthodontists who are lecturing for the companies. A number of orthodontists are also designing and

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promoting their own brackets. How appealing would partial appliances be to those in the business of selling brackets? Bends will be discussed later that offer the choice for extremely effective force systems with larger interbracket distances while shorter interbracket distances limit the use and effectiveness of these bends. Such bends can be easily placed intraorally with a Tweed-Loop pliers and take only seconds. The reward lies in knowledge – not effort – as well as reduced stress, increased efficiency, and greater practice enjoyment for the orthodontist. Figure 4-12 offers an interesting insight to the subject of Forces Forces & moments remain the same. and Relative Magnitudes. Both Only the illustrations show forces that are nomenclature changes. alike when each illustration is “turned around.” In Figure 4Lingual Root Torque is simply a Tip-back Bend “turned around.” 13, an intrusive force is shown on the incisor segment above, Figure 4-12 and an eruptive force on the same segment below. The larger 1c 1a moment on the molar above becomes the smaller moment on the molar below, while the Rectangular Wire The result is smaller moment on the incisor 1b that there are above becomes the larger no vertical moment on the incisor below. Eg. Rectangular or square wire.

forces present.

Does all of this seem to be Figure 4-13 confusing? Simply realize that these two illustrations can represent a single partial appliance

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containing several different activations. In such a case, only equal and opposite moments might occur. Figure 4-14 reveals the net result of combining the force systems seen in the upper and lower portions of Figure 4-13. The significance present in this illustration is that when the tipback bend in an arch is combined with anterior lingual root torque in the same arch, the vertical forces will disappear whenever each of the activations produces Only Equal and Opposite Moments Occur equal and opposite moments. Figure 4-14 In other words, a distal crown moment will take place on the molars while an equal and opposite moment will occur with the incisors. Because the moments will be equal and opposite, there will be no couple necessary for equilibrium. Therefore, when inserting such wire into the molar tubes, the anterior portion of the wire would lie in the muco-labial fold creating the illusion of incisor intrusion. In actuality, when the wire is engaged into the incisor bracket slots, the resulting lingual root moment will eliminate the forces. The orthodontist can wait forever for overbite correction but there can be none if the forces required are not present. This emphasizes the importance of what was taught in Chapter 3, which dealt with the importance of understanding static equilibrium in orthodontics. Making matters even more difficult, if lingual root torque is increased, there will then be a net moment on the system which requires a couple for equilibrium. Thus, an eruptive force will actually exist on the incisor segment although it will still appear

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to the orthodontist that there is an intrusive force when first placing the wire into the molar tubes. This error is created as a result of inserting the archwire into the molar tubes first, as a matter of convenience, and then observing the wire lying in the muco-labial fold. The latter may easily lead an orthodontist to conclude there is an anterior intrusive force when actually an eruptive force takes place upon archwire activation. So, it should not be too difficult at this point to recognize what happens when the net force system involves unequal moments or equal and opposite moments. With the latter, there will be no need for a balancing couple. However, if the moments are unequal, the resulting net moment will require a couple for equilibrium. The result with these outcomes may be an increase in overbite, creation of an open bite, significant occlusal plane changes, as well as other effects that may be considered desirable or undesirable. If the orthodontist has a full understanding of these possibilities, appropriate action may be taken beforehand to prevent undesirable occurrences. While on the subject of forces and relative magnitudes, now is a good Regardless of the time to refer to Figure 4-15, as the applied forces, the net force on a tooth during the resisting forces create a net force application of forces to teeth in of zero. orthodontic treatment is always zero. An intrusive force on an incisor tooth will be opposed by the forces in the periodontium and Figure 4-15 its surroundings. Component forces may be discussed, but the total of all forces acting on a tooth is zero. If a person says he weighs 200 lbs, that is simply the downward component created

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by gravity. There is always an equal and opposite (opposing) force so that the net force will be zero. Again, it is only necessary to recognize that one of the requirements for static equilibrium is that the sum of the forces in a given plane of space must equal zero. In Figure 4-16, it will be seen that The forces in both moments may differ in magnitude planes of space are when vertical and horizontal equal in magnitude. forces are produced. This will be Therefore, the resulting moments a critical element during the later must be unequal. discussion of molar control, as it (F x D = Moment) will be seen that vertical forces Horizontal forces create larger moments. can well be the contributing cause Figure 4-16 of horizontal displacement of the molar crowns due to the moments produced. Because vertical forces usually act buccal to the center of resistance, the perpendicular distance between the vertical force and the center of resistance results in a moment. Eruptive forces will result in lingual crown moments while intrusive forces will create buccal crown moments. When either of these displacements occurs, horizontal forces can be introduced through the molar tubes. It can be seen that horizontal forces acting through the molar tubes will produce larger moments than those produced by the vertical forces, because the perpendicular distance to the center of resistance is larger than the previous. This will be discussed thoroughly under Molar Control, but for now only the concept is presented.

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THE SHORT STORY Forces are frequently measured in orthodontics, whether for patient treatment or for in-vitro studies. When such forces are measured prior to archwire insertion into the bracket slots, a cantilever is involved in such measuring. A cantilever is a simple beam containing a pure force at one end. However, archwires have resiliency which increases with interbracket distance. Therefore, when the wire is fully engaged the resiliency creates wire/bracket angles that did not exist at the time the measurement was taken. The resulting forces are not the same as those which may have been carefully measured. Partial Appliances offer many advantages over Full Appliances, particularly in the initial stages of treatment. The increase in interbracket distance not only increases resilience which can be very effective in producing desirable moments for anchorage and retraction, but it enables the orthodontist to locate bends in such a manner that entirely different force systems may be produced. Small interbracket distances limit this opportunity. Finally, vertical forces acting through molar tubes may cause tooth displacements, but these may be readily corrected by horizontal forces which create larger moments than those causing the displacement. Although the forces in both planes of space are equal because the interbracket distances are the same, the cross-section of the wire is the same, and the activation of the archwire is the same, the moments will differ in magnitude.

Determining Forces and Relative Magnitudes 4

SUGGESTED READINGS Mulligan TF. Common sense mechanics. Phoenix: CSM, 1040 East Osborn Rd, 1982;39-43. Nikolai RJ. Bioengineering analysis of orthodontic mechanics. Philadelphia: Lea & Febiger, 1985;235-237.

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Understanding and Applying Wire/Bracket Angles 5

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Understanding and Applying Wire/Bracket Angles 5 This subject will be the basis for the force systems discussed in the remaining chapters. For many years orthodontists have had to bend archwires to provide force systems of choice. This was time consuming, so the profession welcomed the introduction of prescription brackets to replace the need for such bending. In essence, archwire shape has been traded for bracket design. At the time of this publication there were more than eleven hundred prescription brackets on the market. Think about it! If the ultimate in bracket design, including features such as built-in tip and torque were achieved, then there would be no need for so many different designs. Many problems are associated with such brackets because regardless of the designer, manufacturer, or company, static equilibrium will always require balanced force systems. If the orthodontist is interested primarily in torque at a given time, the entire system must be balanced and even more importantly, recognized. However, when disappointed in a particular type of bracket design, the search goes on for others that are more likely to meet the desired objectives. Obviously, the search continues. Consider other factors as well. In order for a bracket to fulfill the objectives for which it has been advertised, slot sizes and

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cross-sectional dimensions of the archwire must be absolutely accurate, but they are not. In spite of these weaknesses, they do have a place in orthodontics – at least toward the end of treatment when major tooth movements are no longer required. Regardless of whether a prescription bracket is used - or a standard bracket with the so-called neutral slot - the force systems will be formed as a result of the wire/bracket angles created when the archwire is inserted into the bracket slots. These wire/bracket angles can easily be formed in an everyday practice with a Tweed Loop pliers and done so intraorally. The preferable angle formed within the archwire in any given plane of space – frontal, sagittal, or occlusal – is forty-five degrees. This is an angle that is easy to read and can be used effectively with round wire without causing the wire to twist and turn within the slots. Interestingly enough, round wire will be the wire of choice for almost all of the tooth movements discussed, in spite of the bad name given to it over the years because of its inability to produce torque within the slot. The biomechanics involved recognizes the need for torque, but keeping in mind that moments are a product of Force X Distance, it is not necessary to produce torque within the slot. When analyzing the bracket torque produced by square or rectangular wire, it will also be seen that the moment produced is a product of Force X Distance. Because archwires are placed into molar tubes first, as a matter of convenience, the orthodontist experiences the need to twist the wire for anterior bracket engagement. It is assumed that if a wire cannot produce a moment by twisting, it is incapable of producing torque. This is not true.

Understanding and Applying Wire/Bracket Angles 5

Total Force Systems

Forces and Moments

Figure 5-1

56

Lingual Root Torque (F X D = M)

Figure 5-2

Looking at Figure 5-1, it can be seen that bends have been positioned in three different locations. Round wire used in the sagittal plane of space easily creates moments at the brackets because of the angles formed. Keep in mind that certain moments may appear to be shown in the wrong direction, but these moments are correct, as they have occurred due to archwire resiliency which was discussed earlier in great detail. While Figure 5-2 illustrates lingual root torque produced with rectangular wire, it can be seen that the moment is actually a product of Force X Distance, although it is not usually visualized in this manner because of first inserting the wire into the molar tubes. Once inserted into the molar tubes, the wire must be twisted to attain bracket engagement with the anteriors. Round wires are incapable of producing torque within anterior bracket slots because of their round cross-sectional nature. It will later be shown that lingual root torque can be created in the anterior segment with round wire. Keep in mind that intrusive forces acting through the incisor brackets may produce moments when acting labially and at a perpendicular distance to the center

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of resistance in the incisor segment. The moments should really be described as labial crown moments because incisors will flare unless prevented from doing so, at which time the moment will then produce lingual root movement. Before any of this can take place, it is absolutely essential to know the posterior moments. If the posterior moments in a partial appliance are greater in magnitude than the anterior moment, the incisors will actually respond with lingual movement. But this is not the time for such discussion. Wire/Bracket relationships need to be discussed thoroughly prior to introducing the various clinical subjects on tooth movement. WIRE/BRACKET RELATIONSHIPS:

WIRE/BRACKET RELATIONSHIPS Center Bend Off -Center Bend Parallel (Step)

Figure 5-3 provides the names of the three most important wire/bracket relationships to be discussed. There is a fourth one a cantilever - in which a limited discussion will take place because it has already been presented and has limited clinical application.

Figure 5-3 Figure 5-4 simply indicates that with a Tweed Loop pliers, fortyfive degree bends may be placed at different locations specifically chosen by the orthodontist. Each location will produce completely different force systems.

X X X X Different locations of bends produce significantly different force systems.

Figure 5-4

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The locations of the bends – and therefore the force systems – will be determined in the vast majority of cases by choosing one of the locations shown in Figure 5-5. Each location chosen will produce completely different wire/bracket angles. Unlike the X X X prescription bracket, a choice is being made as to which of three Archwires must be loop-free force systems will best solve the We can use any of the the locations marked X to provide problem at hand. It is obvious the force system of choice. that the greater the interbracket distance, the greater will be the Figure 5-5 choices in locating these bends.

NEUTRAL SLOT VS. PRESCRIPTION SLOT

X 1/2 CENTER BEND

IN EACH CASE, IT IS THE WIRE/BRACKET RELATIONSHIP THAT DETERMINES THE FORCE SYSTEM.

Figure 5-6

WIRE-BEND VS. STRAIGHT WIRE

Figure 5-7 THE CENTER BEND Figure 5-6 states that whether the slots are provided by prescription brackets or are simply neutral slots, the wire/bracket angles produced will determine the force system. The original malocclusion will automatically create wire/bracket angles because of malpositioned teeth and a method will be provided later - during clinical treatment - to illustrate resolving or minimizing the side-effects. Figure 5-7 shows the first of

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three wire/bracket angles to be discussed. This one is referred to as the center bend for obvious reasons – the bend is placed equi-distant from the adjacent brackets. All of the angles discussed will be related to the brackets adjacent to the bend in question. Note in this case that the force system produced consists of equal and opposite moments. This is rather obvious to most clinicians, but other systems are not always so obvious.

X 1/2 CENTER BEND

Two Off-Center Bends may be used to produce the same force system as a Center Bend.

Figure 5-8 Figure 5-9 Continuing with the center bend, Figure 5-8 illustrates the resilience that takes place while activating the wire. Resilience is one of the major factors that can often lead to a visual misinterpretation of force systems. The illustration in Figure 5-9 shows that two off-center bends are equal to a center bend. Figure 5-10 involves the paralleling of roots following space closure in four first bicuspid extraction treatment. The application is simple, but Center-bend for paralleling roots significant in response. Figure 5-10

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A diastema is shown in Figure 5-11 with converging Center roots on the central incisors. Bend for A center bend provides the root divergency equal and opposite moments required for root divergency in a Diastema and future stability following closure of the space. Crown movement always proceeds Figure 5-11 root movement until spaces are closed. Further movement then becomes root movement. The next illustration, Figure 5-12, shows that all incisors roots can be diverged at the same time with two sectional wires. A continuous wire is not able to deliver four pure Diverging all roots simultaneously moments without creating vertical forces, but because a center bend is equivalent to Two Off-Center Bends = two off-center bends as seen Center Bend in Figure 5-9, a wire with a Figure 5-12 center bend between the two central incisors and a rectangular by-pass segment with fortyfive degree bends at the lateral incisor brackets, each provide the necessary moments for all four teeth and create root divergency without vertical forces being developed. Using this by-pass wire is one of the very few times a rectangular wire is preferred. Figure 5-13 is another example of using two off-center bends to produce paralleling moments in an extraction site. In this case, a

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military family did not have knowledge as to whether they would remain at Luke Air Force base for a few weeks or for several months. Therefore the decision was made to extract teeth and permit eruption of maxillary cuspids while retracting lower cuspids. Because the mandibular second deciduous molars were still Paralleling in place, the cuspids were Roots retracted with off-center bends placed just mesial to Figure 5-13 the first molar tubes. Remember what was said earlier. The bracket or tube located closest to the bend represents the location of the larger moment, thus making the molars the anchor units during cuspid retraction. However, the cuspid roots then required uprighting so another off-center bend was added as shown in Figure 5-13. This provides the opportunity to parallel roots prior to the eruption of the second bicuspids. Much can be accomplished during the so-called waiting period that might take place while a patient’s family is waiting for a job assignment or a military transfer. Next, the cantilever bend will be discussed. In the center bend, it was seen that the bend was placed exactly in the center between two adjacent brackets. Now the bend will be located at the one-third position between the two adjacent brackets. This results in a long section and a short section with the long section occupying two-thirds of the interbracket distance and the short section covering the remaining one-third.

Understanding and Applying Wire/Bracket Angles 5

X 1/3 OFF-CENTER BEND (CANTILEVER)

WIRE-BEND VS. STRAIGHT WIRE

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X 1/3 OFF-CENTER BEND (CANTILEVER)

Figure 5-14 Figure 5-15 THE CANTILEVER In Figure 5-14 it can again be seen that level brackets with a bend placed in the one-third position produce exactly the same wire/bracket angles as seen when the wire is straight and the brackets are angulated. It now becomes easy to see that determining the total force system by visually interpreting the latter becomes much more difficult. To many orthodontists, the straight wire with angulated brackets may be thought to contain moments at both brackets because of the angles formed when reading an archwire prior to activation. Looking at Figure 5-15, it is seen that when the wire is activated by inserting it into the bracket slots, resilience occurs and results in no angle at the bracket on the right. This is true whether the slot size is .018 or .022 and regardless of whether the interbracket distance is small or large. The rules regarding long section vs short section still exist as the long section points in the direction of the force produced while the short section points opposite in direction to the force produced. The bracket located closest to the bend contains the largest moment and the bracket furthest from the bend contains the smaller moment - or no moment. So the rule itself remains valid but the information given now is more exact, as it describes the smaller moment as being zero.

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Anterior Segment with Continuous Arch Overlay

Figure 5-16 Cantilever intrusion of cuspids is more effective than engaging archwire into bracket slots.

(placed under bracket wing)

Figure 5-17

These are cantilevers for cuspid intrusion.

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Figures 5-16 thru 5-18 illustrate various approaches to the practical application of the cantilever in a clinical practice. Placing the bend in the onethird position as previously discussed is accurate, but small amounts of tooth movement can alter the force system. In Figure 5-16, a continuous archwire overlays an anterior segment. This can be applied in various ways to provide a pure force acting through the center of resistance, if so desired. Finally, Figures 5-17 and 5-18 illustrate the most commonly used cantilever approach for cuspid intrusion. The archwire is placed under the incisal wings of the cuspid brackets and within the slots of the incisor brackets during overbite correction.

Figure 5-18 This concludes the discussion regarding the cantilever bend. Although the cantilever is not required for clinical application, it

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is discussed as one of the several wire/bracket angles. When prescription brackets are being used, this system may very well be encountered with no recognition, as resilience is not visualized when reading wire/bracket relationships. Location of the bends when slots are level provides the most reliable determination of the force systems. Although technically the cantilever just discussed is an offcenter bend, because of the decision to minimize its use, the following wire/bracket angles will be referred to as off-center bends. Otherwise, cantilever bend will be the term used. X OFF- CENTER BEND

X OFF- CENTER BEND

WIRE-BEND VS. STRAIGHT WIRE

Figure 5-19 Figure 5-20 THE OFF-CENTER BEND It will not be necessary to repeat all that was said regarding the previous illustrations as the concept remains the same for each. Figure 5-19 provides a force system quite different than any discussed thus far. For most orthodontists, the smaller moment would appear to be in the wrong direction, but looking at Figure 5-20, it can be seen again that archwire resiliency is responsible for the counterclockwise moment, as the wire crosses the bracket at a completely different angle when activated than when simply visualizing or reading the archwire prior to

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activation as shown in Figure 5-19. The moments appear in the same direction. Forces and moments are predictable by the rules regarding long section vs short section although the smaller moment is more accurately described when the force systems are determined by location of the bends. Instead of referring to the moment as the smaller moment, the specific direction – clockwise, counterclockwise, or zero – is indicated.

Note anterior moment!

Off-Center Bend (Toe-In Bend)

WHY?

Figure 5-21

Figure 5-22

The clinical example shown in Figure 5-21 illustrates the use of a tip-back bend at the molar, which is an off-center bend. As said earlier, round wire cannot produce torque within incisor bracket slots. Although the moments shown previously were in the same direction when occurring in the sagittal plane of space, the anterior moment shown in this figure is opposite in direction because it is formed by an intrusive force acting labial to the center of resistance in the anterior segment, rather than because of torque produced within the incisor bracket slots. The toe-in bend in Figure 5-22 is another form of off-center bend and produces the force and moment on the molar predicted

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by the short section in Figure 5-21. Changing planes of space does not affect the force system on the molar in either figure.

Figure 5-23

Figure 5-24

Figures 5-23 and 5-24 reveal a noncomplicated approach to rotating molars and correcting crossbites simultaneously. The secret lies in avoiding engagement of the archwire into the second bicuspid brackets as seen in the upper arch. If all brackets were engaged, the toe-in (off-center) bends would become center bends and would therefore not provide buccal forces. Remember that the toe-in bend represents the short section and results in a force acting in the opposite direction.

Center Bend utiized to rotate second molar. Second molar now requires buccal movement

Center Bend converted to OffCenter Bend by removing first molar tie. Buccal force is created.

Figure 5-25

It is not uncommon to find molars that have drifted mesially and then rotating and moving into lingual crossbite. This is an easy and effective method for simultaneous correction without the need to remove an archwire.

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Looking at the left side of Figure 5-25, the second molar has been rotated with a center bend. However, it requires some buccal movement, but there is no buccal force present. By simply removing the tie on the first molar, the center bend is converted to an off-center bend, which introduces a buccal force in addition to the molar rotational moment shown. This is certainly a less time-consuming approach than would be required if removing the archwire to place a bend. These are simplified procedures that make orthodontics more fun, less stressful, and probably lead to greater longevity both in practice and life as well. The final clinical example of the off-center bend is shown in Figure 5-26. Locating the bend just mesial to the bicuspid bracket creates an anchor side to the extraction Note Off-Center Bend during site, as the tooth closest to the cuspid retraction. (It will become a Center Bend upon space closure). bend has the largest moment. Once the space is closed, the Figure 5-26 moments become equal and opposite because the bend becomes centered. This change in the force system takes place without removing the archwire at anytime from the very beginning of space closure. Throughout the discussion on wire/bracket angles, it will be noted that brackets are not added or removed for the purpose of creating a particular force system. For example, brackets may or may not be present on cuspids. The various wire/bracket angles may be introduced at any time and do not require adding or removing backets to attain the force system of choice.

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THE PARALLEL (STEP) BEND X X PARALLEL (STEP) BEND

X

X

PARALLEL (STEP) BEND

WIRE-BEND VS. STRAIGHT WIRE

Figure 5-27

Figure 5-28

The final wire/bracket relationship to be introduced is the parallel relationship - also referred to as the step. In Figure 5-27 the same two situations are repeated as shown in the previous. It is obvious again that the wire/bracket angles are equal whether the wire contains activating bends and level slots, or involves a straight wire with angulated slots. The step bend, when compared to the off-center bend shown previously, is nothing more than an additional bend placed at the adjacent bracket and reversed in direction, thus creating two parallel short sections. Of all the wire/bracket angles discussed, this relationship involves the highest force magnitudes. The reason is very obvious. Since both moments are equal in magnitude and act in the same direction, the net moment is the highest of any demonstrated thus far. Therefore the balancing forces must be greater in order to create a balancing moment, a moment which must be equal and opposite to the net moment just discussed. The step relationship, because of the higher force magnitudes, will find its greatest usefulness in the occlusal plane of space when positioning molars that have been displaced such as in

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crossbites. The higher force magnitudes may be desirable in the treatment of older individuals where response is usually not as great as that seen with younger patients. The step bend can also be applied in the sagittal plane of space when vertical forces do not pose a threat to the vertical dimension of the patient. Brachycephalic individuals provide good examples.

Cuspids bonded

Cuspids not bonded

Parallel (Step) Bend

Figure 5-29

When molar and cuspids are bracketed, a true parallel wire/bracket relationship exists.

(Rigidity vs Resilience)

Figure 5-30 / Figure 5-31

Figure 5-29 shows two examples of the step bend. In the left picture the cuspid brackets are present while absent in the picture to the right. Technically speaking, the first picture is a genuine step relationship while in the second, bends are applied in the same manner, but with cuspids not bonded. As a result of cuspid brackets not being present, a true step relationship does not exist and the forces produced will be less in magnitude. However, this is a practical approach as will be seen later in the discussion on molar control. The higher force magnitudes in Figure 5-30 and 5-31 result from a step relationship and are very effective in restoring central groove relationships between first and second molars and increasing the posterior transverse dimension for the patient.

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The step bend is only used when higher force magnitudes are desired. Otherwise, the off-center bend is the bend of choice. Because of the high forces that can be created with the step bend, it must be recognized that it can have distinct disadvantages in the sagittal plane of space when there are vertical dimension problems present. This is particularly Parallel (Step) Bends true with full appliances and in Figure 5-32 many cases the use of the step bends as seen in Figure 5-32 has led to the need for posterior high-pull headgear in order to overcome extrusive movements. The step bend creates force magnitudes that are up to four times greater than those produced by a cantilever acting at the same interbracket distance with the same degree of activation. In closing this chapter, a further discussion of the step bend and the cantilever will be presented in order to better understand each.

X

What happens if we section the parallel bend?

ANSWER: We recognize that the parallel bend is actually a combination of two half-length cantilevers.

Figure 5-33 Figure 5-34 In Figure 5-33, it is decided to section a step bend into two halves. In Figure 5-34, the two halves constitute cantilevers.

Understanding and Applying Wire/Bracket Angles 5

LOAD/DEFLECTION RATE FOR CANTILEVERS The load/deflection rate is inversely proportional to the cube of the wire length. If the length is doubled, the force becomes 1/8 per unit of deflection. Therefore, if we reduce the length to one-half, the force will increase 8X. (Assuming vertical deflection of the archwire remains constant)

Figure 5-35 Figure 5-36 illustrates the stepdown arch. An increase in archwire length does not result in an increase in the vertical deflection. The load/deflection rate is 1/8th the force per unit of deflection. If the length is doubled, it will take 1/8th as much force for the same wire deflection. In Figure 5-37, a tip-back bend is shown. When the wire length is doubled with this type of wire activation, vertical deflection will also double. If the force is reduced to 1/8th per unit of deflection and there are now two units of deflection, then 2 x 1/8th = 1/4th of the original force.

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Figure 5-35 explains the load/deflection rate for cantilever beams and is useful in helping to explain why force levels are as high as previously stated. Keep in mind the archwire activation may involve different bends as seen in the figures below.

LOAD/DEFLECTION RATE Step-down arch

As the length of the wire doubles, it takes only 1/8th the force per unit of deflection. (Step-down arch)

Figure 5-36 LOAD/DEFLECTION RATE Tipback arch

But with the tip-back bend, when the length doubles there are two units of deflection requiring twice the force. (2x1/8) = 1/4th as much force.

Figure 5-37

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Since the tip-back bend (off-center bend) will be the activation of choice throughout this book, Figure 5-37 will be the one that will be most important to understand and remember.

One Unit of Deflection X Two Units of Deflection

4X

1/8 X

1/2 X

Force = X

Force = X

X

1/4 X

Figure 5-38

Figure 5-39

Figure 5-40

Figure 5-38 shows the increasing vertical deflection with the use of the tip-back bend. Because the vertical deflection doubles when the wire length is doubled, the load/deflection rate shown in Figure 5-39 is 1/4th the original. Figure 5-40 reveals that a half-length cantilever will require four times as much force for activation than will the same cantilever with twice the length and only one unit of vertical deflection. Finally, this subject will close with comments regarding the cantilever. It seems that the cantilever has created much confusion with many orthodontists regarding the fact that no moment is present at a bracket where one would appear to be based on visual inspection of that bracket. Because reading an archwire prior to insertion into the bracket slot indicates the presence of an angle, it is easy to assume that a moment must be present at that bracket. Although an attempt has already been made to clarify the issue, the questions do not seem to disappear. In order to further clarify the matter beyond the explanations already given regarding resilience, an additional explanation will be offered which should eliminate any doubts.

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Center Bend

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Understanding the Cantilever

Observe the right bracket

Step Bend

Figure 5-41 Understanding the Cantilever

In order to go from clockwise to counterclockwise, the transition must encounter zero.

As the bend moves to the left, the wire/bracket angle on the right continues to reduce in a counterclockwise direction.

Figure 5-42 Understanding the Cantilever

In order to go from clockwise to counterclockwise, the transition must encounter zero.

Figure 5-43 Figure 5-44 When the various wire/bracket angles were discussed, the extremes involved the center bend with equal and opposite moments and the step bend with equal moments in the same direction. The latter also had balancing forces present while none were required in the center bend. Looking at Figure 5-41, note that the brackets shown on the right involve opposite angular relationships. In Figure 5-42, it is seen that as the bend is moved from the centered position to the final step position, the new angles formed have all rotated counterclockwise. In Figures 5-43 and 5-44, the bracket slot has moved from its original angular position to its final angular position, which is now opposite the original. It should now be evident that a wire

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cannot rotate from one angle to an opposite angle without passing through a zero angle. The zero point is shown in Figure 5-45 and represents a point at which only a pure force is present at the bracket. Note the archwire forming no angle with the slot. Understanding the Cantilever X 1/3

In order to go from clockwise to counterclockwise, the transition must encounter zero.

Figure 5-45

CANTILEVER

Figure 5-46

Figure 5-46 is shown once again to illustrate that archwire resiliency is responsible for the zero angular relationship whenever a bend is placed at the one-third position between two brackets. These statements apply to loop-free wires. For clarification purposes, resiliency causes a zero angle in the onethird position and this zero angle must exist somewhere between the two extremes of a clockwise moment in the center bend and a counterclockwise moment in the step as seen in Figure 5-41. Nothing in life happens without a reason and orthodontics is part of our lives. Understanding the causes - and not just the effects affords the opportunity to add innovative treatment procedures to an everyday orthodontic practice. You just might decide to add years to your practice and life!

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THE SHORT STORY There are several wire/bracket relationships that may take place in a loop-free wire which offer the orthodontist an opportunity to choose specific force systems by simply choosing a particular location for placing an activational bend. Three of these angular relationships are extremely useful in orthodontic treatment, particularly at a stage where partial appliances are in use. Although a cantilever system may be created by properly locating a bend, there are more practical ways to create such force systems in an orthodontic practice. The archwires may be placed beneath the wings of the brackets rather than in the slots. Center bends may be very useful for paralleling roots following space closure and diverging roots in diastemas, as well as being beneficial in several other areas of treatment. The off-center bends are extremely effective for retracting and protracting teeth, for buccal and lingual movements, rotational corrections, vertical movements, and several other movements, as well. The step bend produces the highest force magnitude of any of the wire/bracket relationships and has its particular usefulness in the occlusal plane of space. Because it produces equal moments in the same direction, the balancing forces are higher than the other relationships and can therefore be a threat to vertical dimension problems when used in the sagittal plane of space. It is extremely effective for increasing posterior transverse dimensions and is the relationship of choice for the older patient.

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SUGGESTED READINGS Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J Orthod 1974;65:270-289. Burstone CJ. Rationale of the segmented arch. 1962;48:805-822.

Am J Orthod

Mulligan TF. Common sense mechanics office course. Phoenix, AZ.