16 M. Kirschneck et al. Rayleigh damping was used to approximate the damping behavior of the beam. The coefficients of the Rayleigh damping˛ andˇwere determined by measuring the damping ratio of the system without any interaction with the magnetic field. It was assumed that the damping ratio is the same for 417 Hz and 370 Hz. The Rayleigh damping coefficients were tuned in a 3D FEM model without any magnetic coupling until the measured damping ratio for the two frequencies was reached. Du D˛Mu CˇKuu (2.8) The magnetic stiffness matrix was calculated in the same way as the mechanical stiffness matrix. Instead of the potential energy the magnetic energy was used. KAA D @2Wmag @q2 A Z .r A/T 1 .r A/d (2.9) the magnetic mass matrixMA can be calculated out of the source term @A @t using the Galerkin method. MA D @2 @qA@PqA Z AT PAd (2.10) For the calculation of the magnetic force vector Fmag the Maxwell stress tensor T was used [11,12,14]. This tensor is derived from the principle of virtual work that can be derived from (2.3). It can be shown analog to [10] that the magnetic force can be calculated by Fmag D @ @qu Z .r A/T 1 .r A/d (2.11) The magnetic force represents the first coupling (in this case from the magnetic domain to the mechanical domain). Due to the distortion of the domain caused by the displacement u the magnetic stiffness and mass matrix as well as the magnetic force vector depend on the displacement u. For the magnetic stiffness and mass matrix this dependency is crucial because this dependency will cause a coupling between the mechanical displacement and the magnetic field. With these matrices we can formulate the non linear set of equations that describe the coupled system. Muu Rqu CDuu Pqu CKuu qu DFext CFmag.qu; qA/ (2.12) MA.qu/ PqA CKAA.qu/ qA DJext C 1 .r Br/ Each physics for themselves is linear. Coupling the two physical domains will cause the complete set of PDEs to become non linear. Therefore in order to do a modal analysis they need to be linearized. As the oscillation of interest is around the undeformed configuration of the structure and the static magnetic field generated by the permanent magnets, the linearization point is given by those to states. For such a linearization point (2.12) can be transformed into a linear monolithic system of equations: Mu 0 0 0 „ ƒ‚ … M Rqu RqA C Du 0 0 MA „ ƒ‚ … D Pqu PqA C Kuu KuA KAu KAA „ ƒ‚ … K qu qA „ƒ‚… q D Fext Jext CLPM „ ƒ‚ … L (2.13) The coupling matrices KuA and KAu can be derived from the magnetic force and therefore from the energy stored in the magnetic field. KuA D @Fmag @qA D @ @qA @ @qu Wmag D @ @qu @ @qA Wmag D @ @qu .Jint / DKAu (2.14) Looking at Eq. (2.14) it can be assumed that the total stiffness matrix is symmetric. However, in the FEM code used for the simulation the stiffness coupling matrixKuA is derived in another way which ruins this symmetry. Using the Maxwell stress tensor Tthe specific magnetic force acting on a structure can be computed by fmag Dr TDr HBT I H B (2.15)
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