Abstract: The invention is a mounting system for a vibrational element. The vibrational element has a longitudinal axis, an outer surface and an axial displacement node. The mounting system includes an isolation member which has an isolation surface, an isolating body, and an engaging portion. The engaging portion is immovably secured to the isolation member and extends radially inward from the isolation body at a point radially opposite the isolation surface. The engaging portion is disposed so as to engage the outer surface of the vibrational element. Radial displacements of the vibration element are substantially decoupled from the isolation surface through the engaging portion.

Abstract: Methods and apparatus for orthodontic treatment planning that involve determining an initial position of an orthodontic structure, determining an altered position, and generating response surface data using a mathematical relationship between the positions. The mathematical relationship may be defined by a number of parameters useful for determining one or more of the stress, strain, force, and moment associated with movement of the orthodontic structure. The mathematical relationship between the initial position and the altered position may be defined using a finite element analysis, empirically determined, or using other computational methodologies such as a finite difference methodology. The mathematical relationship may also be defined using an analytical methodology, such as elasticity and/or plasticity methodologies.