In our last post (Part 1: Clarifying terms), we explained the Newton’s laws applied to orthodontics, with emphasis on the first law, THE LAW OF INERTIA, by which we showed that ALL orthodontic appliances must be in STATIC EQUILIBRIUM when installed. But do you know exactly what it means to be in balance? Do you know how to assess and recognize this condition? And do you know the clinical applications of this knowledge?

These are the questions that we want to clarify in this post. Being in static equilibrium means not moving, or more specifically, not accelerating. If you install a spring, or elastic, or any other device capable of generating force, certainly this device will not suffer acceleration. The forces generated by the appliances originate from the elastic deformation of its components, which tend to return to their original shape after their interatomic bonds have changed. This elastic deformation, however, does not represent an acceleration. When installing an elastic stretched between two teeth, for example, the elastic will not move to the right, nor to the left, nor up, nor down. That is, the appliance (in this case the elastic) will be in equilibrium. We can easily visualize the forces that hold it in this condition: they are the equal and opposite forces that the brackets exert on the elastic. As we explained in the previous post, these are the activation forces, which will trigger forces of reaction (or deactivation) responsible for tooth movement (Figure 1).


FIGURE 1. A. Elastic shape before force application. B. After applying two equal and opposite activation forces (blue), note lack of motion relative to a fixed reference point (x). C. reaction/deactivation forces (red) acting on the brackets.


And how to evaluate this equilibrium condition? Our only task will be to RECOGNIZE the systems of forces involved in each particular situation. That is, we should be able to visualize the forces and moments necessary to establish the state of equilibrium. In this state, TWO requirements must be met:

1) The sum of all the forces (vertical and / or horizontal) present must be zero.

2) The sum of the moments acting at ANY point must also be zero.

In the example of figure 1, we easily find the equilibrium of the horizontal forces. As they are collinear (have coincident lines of action), the cancel each other naturally without producing any rotation (or moment) in the appliance. In this simple example, the requirements for equilibrium were easily met and recognized.

However, when we apply vertical and / or non-collinear forces in a system, there are rotational forces that are not always easily visualized. Understanding the principle of equilibrium will help us in identifying the forces and moments present in any appliance.

To facilitate its study, we will explain 4 different situations, which may occur between the infinite possibilities of misalignment between two teeth. In our examples, we will use 2 premolars.

Situation 1) Two teeth are rotated in opposite directions, but in the same proportion relative to the interbracket axis. To insert a straight wire between these brackets, you will need to do two equal and opposite moments to elastically deform the wire. These moments (of activation) represent the actions of the brackets on the wire, and are obviously in equilibrium, as stated in Newton’s first law.


Figure 2. Situation 1) A. Two activation couples, in blue, are required to insert a straight wire (wire in solid line has been elastically deformed to the red dashed line) between the brackets. B. Deactivation-reaction forces, in red-that will be perceived by the teeth. This situation represents a class VI geometry.

Situation 2) Two teeth are rotated in opposite directions, one being rotated to half the quantity of the other, in relation to the interbracket axis. In this case, given the equilibrium principle, the forces and moments required to insert a straight wire in these brackets consist of an counterclockwise moment and an intrusive force at the end of the wire on the side of the tooth b, while only one extrusive force is applied to the end of the wire on the side of the tooth a. Note that the moment of side b(counterclockwise) will be counterbalanced by the moment generated by the two forces acting on the ends of the wire. That is, the couple produced by the vertical forces in a and b is clockwise, and thus neutralizes the moment of side b. Remember the second requirement of equilibrium: the sum of the moments at ANY point in the system must be zero. If you choose a point (on tooth a, tooth b, or any other point on the wire) to calculate these values, you will verify this requirement. For practical learning purposes, we suggest that you do not worry about values ​​at this time. Just train your “clinical eye” to visualize the balance of the system.


Figure 3. Situation 2) A. One couple on end B and one couple across the length of the wire, in blue, are required to insert a straight wire (wire in solid line has been elastically deformed to the red dashed line) between the brackets. B. Deactivation-reaction forces, in red-that will be perceived by the teeth. This situation represents a class IV geometry.

Situation 3) Whenever you need to perform opposite moments to insert a wire between two brackets, being one moment longer than the other, there must certainly be vertical forces at the ends of the wire for the principle of equilibrium to be met. This situation occurs, for example, if two teeth are rotated in opposite directions, one being rotated three quarters the quantity of the other in relation to the interbracket axis. Notice again how the equilibrium has been reached: the greatest counterclockwise moment of the side b was counterbalanced by the smallest moment of the side a plus the moment generated by the vertical forces of the extremities (they form a clockwise couple).


Figure 4. Situation 3) A. In this case, three couples are required for equilibrium. Two couples of opposite directions and different magnitudes act at the ends, while one couple along the wire-activation forces, in blue- keeps the system in equilibrium. B. Deactivation/reaction forces, in red, that will be perceived by the teeth. This situation represents a class V geometry.

Situation 4) Finally, there are situations in which two equal moments in the same direction are necessary to insert a wire between two brackets. This occurs, for example, when the two teeth are rotated in the same direction, and in the same proportion with respect to the interbracket axis. In this case, as always, the equilibrium has been reached, since the moments at each end (the two counterclockwise) will be counterbalanced by the moment acting along the length of the wire (ie. the clockwise couple generated by the vertical forces in the ends of the wire). Some variations of this situation occur when the moments of the extremities are in opposite directions and have different intensities. In these cases, these moments will also be counterbalanced by a couple generated by the forces of the extremities, the only difference being in the intensity of these forces, which will be smaller than in the illustrated example of the situation 4.


Figure 5. Situation 4) A. In this case, two couples with the same direction and the same magnitude acting at the ends are balanced by one couple acting across the wire-activation forces, in blue. B. Deactivation/reaction forces, in red, that will be perceived by the teeth. This situation represents a class I geometry.


How to apply this knowledge clinically?

The answer to this question requires a more in-depth study of the subject, but the understanding of the principle of equilibrium represents one of the first steps for the rational and effective application of biomechanical principles in orthodontic clinic.

The beginner may even classify this knowledge as too theoretical and complex, though it forms the basis for understanding how the appliances work. Applying the theory of equilibrium in the day-to-day clinical practice will bring innumerable positive surprises to the professional. You will be able to simulate and predict the desired dental movements in order to select and build the appliances with more consistent force systems. You will notice that sometimes some movements are scientifically impossible to realize with just one device or activation, because we will never be able to disrespect the laws of physics (eg. create an appliance that is not in equilibrium). In addition, you will be able to assess the actual needs of anchorage control in order to minimize undesirable side effects, while potentiating the desired movements.

TIP: The best way to study and apply the principle of equilibrium is by drawing “equilibrium diagrams”, in which you draw the desired force systems and check if it is in equilibrium. Remember that the desired forces and moments (the deactivation ones) are exactly opposite the forces and moments of activation, which ALWAYS must be in equilibrium. Therefore, if your force system is in equilibrium, there will be the possibility of applying it at the clinic. Certainly, your system will fit into one of the four situations described in this post. In the future, we will explain other ways of naming and classifying these situations (the Burstone geometries), and we will describe some of the infinite possibilities of applying this knowledge to the confection of precise and efficient appliances. THE GOOD NEWS is that YOU will be able to create the desired configuration (eg geometry) with simple bends in a great number of appliances.

We conclude by inviting the colleagues to learn more about these biomechanical principles (see recommended reading). For those who haven´t had the time to go deep into complex scientific articles, we also suggest our online course* – focused on teaching the fundamental concepts of bone biology and orthodontic mechanics.

*Enjoy our 10% discount on the Online Course Introduction to Orthodontic Biomechanics and reach this knowledge with updated and truly didactic material!

Recommended Reading:

  1. Burstone CJ, Koenig HA. Force systems from an ideal arch. Am J Orthod. 1974;65:270-89.
  2. Mulligan TF. Common sense mechanics-static equilibrium. J Clin Orthod 1979;13:762-6.
  3. Shellhart WC. Equilibrium clarified. Am J Orthod Dentofacial Orthop. 1995;108:394-401.
  4. Sakima MT, Dalstra M, Loiola AV, Gameiro GH. Quantification of the force systems delivered by transpalatal arches activated in the six Burstone geometries. Angle Orthod. 2017 Jul;87(4):542-548.