Proceedings of the ASME 2013 International Mechanical Engineering Congress & Exposition IMECE2013 November 15-21, 2013, San Diego, California, USA IMECE2013-66010 AN ANALYTICAL MODEL OF MICROMACHINED ELECTROMAGNETIC INDUCTIVE CONTACTLESS SUSPENSION Kirill V. Poletkin * Department of Microsystems Engineering - IMTEK University of Freiburg Georges-K ¨ ohler-Allee 103 D-79110 Freiburg, Germany Email: kpoletkin.edu@gmail.com Christopher Shearwood The School of Mechanical & Aerospace Engineering Nanyang Technological University 50 Nanayang Avenue, 639798 Singapore Email: mcshearwood@ntu.edu.sg Alexsandr I. Chernomorsky The Department of Automatic Complexes of Orientation and Navigation Systems Moscow Aviation Institute 125993 Moscow, GSP-3, Russia Email: chernomorscky@yandex.ru Ulrike Wallrabe Department of Microsystems Engineering - IMTEK University of Freiburg Georges-K ¨ ohler-Allee 103 D-79110 Freiburg, Germany Email: wallrabe@imtek.uni-freiburg.de ABSTRACT The paper presents an analytical model of a micromachined electromagnetic inductive contactless suspension, which de- scribes the dynamics of a levitated disk shaped proof mass in space, near an equilibrium point. The proof mass is levitated in an electromagnetic field created by a ring shaped coil. The model derives from the analysis of the set of Lagrange - Maxwell equations obtained for the proof mass - coil system in a general form. Also the condition for the stable levitation of the proof mass in space is developed and expressed in terms of coefficients of the quadratic form of a function of mutual inductance between the disk shaped proof mass and ring shaped coil. Keywords: levitation; micromachined inertial sensors; con- tactless suspension NOMENCLATURE F l , F s generalized forces ( N ) * Corresponding author. ~ F ext vector of external force ( N ) ~ F p vector of ponderomotive force ( N ) g gravity acceleration ( m·s -2 ) h height of proof mass levitation ( m ) i 1 , i 2 currents ( A ) J moment of inertia ( kg·m 2 ) ~ K vector of angular momentum ( kg·m 2 ·s -1 ) l generalized coordinate ( m ) L Lagrange function ( J ) L 1 self inductance of coil ( H ) L 2 self inductance of proof mass ( H ) L m mutual inductance between coil and proof mass ( H ) m mass ( kg ) M ϕ generalized torque ( N·m ) ~ M ext vector of external torque ( N·m ) ~ M p vector of ponderomotive torque ( N·m ) P stiffness matrix ~ r A ,~ r B ,~ r O vectors of position ( m ) R 1 resistance of coil ( Ω ) 1 Copyright c  2013 by ASME