Author's response

Abstract

I do not consider myself an expert in mechanics, and I was pleased that Dr Burrow has a background in engineering, an immense asset for understanding the principles of biomechanics. Our different backgrounds might explain his confusion on my using the term “classical mechanics.”The relevant Wikidepia page (http://en.wikipedia.org/wiki/Mechanics, accessed June 4, 2008) defines mechanics as “the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements.” Further down, it says, “The major division of the mechanics discipline separates classical mechanics from quantum mechanics,” and still further, “The following are described as forming Classical mechanics: Newtonian mechanics, … Statics, … Biomechanics.” In any case, all this is beside the main argument.I agree with Dr Burrow that a lot of the confusion arises because of inadequate definition of the terms used. The true phenomena can be modeled at different levels of complexity, from the simple mechanical 2-dimensional viewpoint that we have been discussing to a more complex 3-dimensional level that takes into account deformation of the materials (a finite element approach would be indicated here) to an even more complex level that includes the biologic processes. The assumptions and the validity of each model need careful consideration to reliably answer the question of whether the 2 teeth will move by equal amounts.Dr Burrow's explanation of the forces and moments (taking tipping into account) are not in doubt. However, the fact that friction might be greater at 1 of the 2 brackets does not imply that a tooth will move less. Let me repeat the initial conditions to avoid misunderstandings: 2 identical teeth, different-width brackets, joined by a straight piece of wire (that might deflect a bit), the wire being a sectional piece (not extending to any other teeth), and an elastic force between the teeth at the bracket level. The whole system just described (after the teeth have tipped and the wire deflected) is in a state of static equilibrium; otherwise, it would accelerate out of the mouth according to Newton's laws. Therefore, the total external force on the system must be zero. The only source of external force is the periodontal support at the 2 identical roots. It follows that the resultant forces on the roots should be equal and opposite (to give a zero sum); therefore, the biologic response should be the same, and tooth movement should be the same. There are many simplifying assumptions in this argument—eg, that tipping of the teeth and wire deflection are small. Otherwise, we would expect vertical forces, but these assumptions do not invalidate the argument.Equal tooth movement seems to be contradicted by the unequal frictional forces demonstrated by Dr Burrow. However, this is not the case because the wire is a sectional piece. It makes no difference what the relative friction at each bracket is; the wire will bind at 1 bracket and slide at the other (the lowest friction bracket), so, effectively, there is only 1 frictional force. Even if we solder the wire at 1 bracket, thus creating “infinite friction,” both teeth will still move by equal amounts. I repeat the caveat: the wire is a sectional piece, not engaging any other tooth. If this does not hold, then, as I explained in my previous letter (Halazonetis DJ. Friction and anchorage loading. Am J Orthod Dentofacial Orthop 2008;133:484-5), and in agreement with Dr Burrow, frictional forces at each bracket do become important in dictating the relative movement of each tooth.I think that our apparent disagreement stems from an inadequate statement of the initial conditions, and I would like to thank Dr Burrow for his expert viewpoint. I do not consider myself an expert in mechanics, and I was pleased that Dr Burrow has a background in engineering, an immense asset for understanding the principles of biomechanics. Our different backgrounds might explain his confusion on my using the term “classical mechanics.” The relevant Wikidepia page (http://en.wikipedia.org/wiki/Mechanics, accessed June 4, 2008) defines mechanics as “the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements.” Further down, it says, “The major division of the mechanics discipline separates classical mechanics from quantum mechanics,” and still further, “The following are described as forming Classical mechanics: Newtonian mechanics, … Statics, … Biomechanics.” In any case, all this is beside the main argument. I agree with Dr Burrow that a lot of the confusion arises because of inadequate definition of the terms used. The true phenomena can be modeled at different levels of complexity, from the simple mechanical 2-dimensional viewpoint that we have been discussing to a more complex 3-dimensional level that takes into account deformation of the materials (a finite element approach would be indicated here) to an even more complex level that includes the biologic processes. The assumptions and the validity of each model need careful consideration to reliably answer the question of whether the 2 teeth will move by equal amounts. Dr Burrow's explanation of the forces and moments (taking tipping into account) are not in doubt. However, the fact that friction might be greater at 1 of the 2 brackets does not imply that a tooth will move less. Let me repeat the initial conditions to avoid misunderstandings: 2 identical teeth, different-width brackets, joined by a straight piece of wire (that might deflect a bit), the wire being a sectional piece (not extending to any other teeth), and an elastic force between the teeth at the bracket level. The whole system just described (after the teeth have tipped and the wire deflected) is in a state of static equilibrium; otherwise, it would accelerate out of the mouth according to Newton's laws. Therefore, the total external force on the system must be zero. The only source of external force is the periodontal support at the 2 identical roots. It follows that the resultant forces on the roots should be equal and opposite (to give a zero sum); therefore, the biologic response should be the same, and tooth movement should be the same. There are many simplifying assumptions in this argument—eg, that tipping of the teeth and wire deflection are small. Otherwise, we would expect vertical forces, but these assumptions do not invalidate the argument. Equal tooth movement seems to be contradicted by the unequal frictional forces demonstrated by Dr Burrow. However, this is not the case because the wire is a sectional piece. It makes no difference what the relative friction at each bracket is; the wire will bind at 1 bracket and slide at the other (the lowest friction bracket), so, effectively, there is only 1 frictional force. Even if we solder the wire at 1 bracket, thus creating “infinite friction,” both teeth will still move by equal amounts. I repeat the caveat: the wire is a sectional piece, not engaging any other tooth. If this does not hold, then, as I explained in my previous letter (Halazonetis DJ. Friction and anchorage loading. Am J Orthod Dentofacial Orthop 2008;133:484-5), and in agreement with Dr Burrow, frictional forces at each bracket do become important in dictating the relative movement of each tooth. I think that our apparent disagreement stems from an inadequate statement of the initial conditions, and I would like to thank Dr Burrow for his expert viewpoint. Friction and anchorage loadingAmerican Journal of Orthodontics and Dentofacial OrthopedicsVol. 134Issue 2PreviewDr Halazonetis's recent letter to editor (Am J Orthod Dentofacial Orthop 2008;133:484-5) entitled “Friction and anchorage loading” certainly made frictional forces seem simple; he referred to “classical mechanics” twice, and I must admit that term is confusing. When I was in engineering school, we studied statics, dynamics, thermodynamics, and so on, but never classical mechanics. I think that part of the difference of opinion in this controversy is that no one has defined friction in real terms and its relationship to resistance to sliding. Full-Text PDF