### Part 3: Quandles, Inverting Triangles to Triangles and Inverting into Concentric Circles

Jaime Rangel-Mondragón

Published August 24, 2017

This article continues the presentation of a variety of applications around the theme of inversion: quandles, inversion of one circle into another and inverting a pair of circles into congruent or concentric circles. Read More »

Desmond Adair, Martin Jaeger

Published June 15, 2017

Input shaping is an established technique to generate prefilters so that flexible mechanical systems move with minimal residual vibration. Many examples of such systems are found in engineering—for example, space structures, robots, cranes and so on. The problem of vibration control is serious when precise motion is required in the presence of structural flexibility. In a wind turbine blade, untreated flapwise vibrations may reduce the life of the blade and unexpected vibrations can spread to the supporting structure. This article investigates one of the tools available to control vibrations within flexible mechanical systems using the input shaping technique. Read More »

Zachary H. Levine, J. J. Curry

Published March 28, 2017

The derivation of the scattering force and the gradient force on a spherical particle due to an electromagnetic wave often invokes the Clausius–Mossotti factor, based on an ad hoc physical model. In this article, we derive the expressions including the Clausius–Mossotti factor directly from the fundamental equations of classical electromagnetism. Starting from an analytic expression for the force on a spherical particle in a vacuum using the Maxwell stress tensor, as well as the Mie solution for the response of dielectric particles to an electromagnetic plane wave, we derive the scattering and gradient forces. In both cases, the Clausius–Mossotti factor arises rigorously from the derivation without any physical argumentation. The limits agree with expressions in the literature. Read More »