2 Effects of Magneto-Mechanical Coupling on Structural Modal Parameters 13 The force acting on the beam is therefore smaller while it moves away from the equilibrium position than when it moves towards it. This slows the oscillation down resulting in an increased damping. However the force is not linearly dependent on the velocity of the beam. This distorts the oscillation behavior of the beam slightly. 2.2.2 Mathematical Description of the System The system consists of two domains: the mechanical structural dynamics of the beam and the magnetic field. It is therefore necessary to use a coupled model of these two physics to describe it completely [6]. The mechanical system can be described as a second order system Rui C @ ij @xj Cfi;ext D0 (2.1) where denotes the stress tensor and fext the external force. The magnetic field can be described by the magnetic vector potential A r 1 r AD @A @t C 1 r Br (2.2) where represents the permeability of the material, Br the remanence flux density of the magnets and the conductivity of the material. Both parts of the system can store energy. Assuming a conservative system the energy between the mechanical system and the magnetic domain can be exchanged in both directions. The total energy in the system can therefore be calculated by W DWmech CWmag DWkin CWpot CWmag This can be seen as a potential energy for small displacements (the magnetic potential is not defined for all points in the domain due to singularities at corners. However as long as the integration path does not encircle such a singularity the energy is conservative.) As stated in [5] the change of the magnetic field energy can be described by dWmag Did Cfmagdu (2.3) where is the flux linkage, i the currents in any eventual coils, fmag the magnetic force and u the displacement. In this system however we can ignore the first term on the right hand side as there are no coils present. Extending this kind of analysis to the whole system it can be concluded that the only ways of energy entering or leaving the system is by means of external forces fext , coils, mechanical friction and ohmic losses. dW Dd PuC ieddy Cicoil d Cfext du (2.4) the above mentioned energy exchange by forcefmag dubecomes in this case an internal energy conversion from the magnetic domain to the mechanical domain and vis versa. It can be seen from (2.3) the magnetic force can be calculated using the principle of virtual work fmag D @Wmag @u (2.5) 2.2.3 Parameter Identification In order to determine the magnet properties of the steal used for beam and yoke impedance measurements were conducted. The permeability of metals depends on the manufacturing process. Therefore it is hard to predict this property beforehand. However, this property can be determined by measuring the impedance of a coil winded around the beam or the yoke. This property depends mainly on the conductivity of the material and the permeability. For structural steal that is used in this case the conductivity is roughly known. Therefore the impedance can be used to approximate the permeability. By simulating the same system in a 3D FEM program the permeability of the material can be estimated. Figure 2.2 shows the comparison between the measured values for the inductance and resistance and the calculated values for different permeabilities and conductivities of the iron material. The instrument used was lacking the capability to measure below a frequency of 20 Hz. It is presumed that due to the skin effect in the iron the inductance drops rapidly for some frequencies
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