12 M. Kirschneck et al. Recent research on dynamics of magneto-mechanical coupled system concentrated on one way coupled formulations [4, 8, 9]. The reason for this is that 3D magnetic calculations are expensive and are avoided unless absolutely necessary. Research on 3D two way coupled problems has been done but not applied to modal analysis [2]. 2.2 The Test Setup The test set up consists of a stator yoke, two permanent magnets and a flexible beam. Figure 2.1 shows a photo of the test set up and a 3D schematic of it. The coil that can be seen in the picture was not used for the experiments. The stator yoke is fixed to the table by clamps. The flexible beam is fixed to a table that can be moved. This construction allows to move the front part of the beam in and out of the air gap of the stator yoke. The two permanent magnets are located in the air gap and create the magnetic field that passively interacts with the structural dynamics (Table 2.1). Because neither the stator yoke nor the beam are slotted eddy currents are possible in the system and heat dissipation can occur. 2.2.1 Emerging Effects The beam is constructed in such a way that the first bending frequency in one direction is much lower than in the other directions. The bending mode shown in Fig. 2.1b will decrease the air gap length on one side of the beam while it is increased on the other side of the beam. This will change the magnetic field in the air gap. Due to fringe effects the magnetic flux density will rise on the side where the air gap length is reduced and diminish where the air gap lengthened. The resulting magnetic force that acts on the beam and pulls the beam in both air gaps towards the yoke will also change. Because the force does not depend linearly on the air gap length but is proportional to 1=l, where l is the air gap length, the forces will no longer even each other out and the beam will see a force pulling it in the same direction as the displacement. From a dynamical point of view this can be seen as an additional negative stiffness that is introduced into the system when the beam oscillates. As a result the oscillation frequency of the first bending mode will decrease. Additionally the time changing magnetic field will induce eddy currents in the stator yoke counter acting the change of the magnetic field. The result is that the peak of the magnetic field has a short delay compared to the peak of the displacement. permanent magnet beam yoke 1st Bending Mode Fig. 2.1 The test rig used for measurements Table 2.1 Specification of permanent magnets as documented by the suppliers (if documented) Property Value Height 2 mm Length 20 mm Width 10 mm Remanence flux density 1.32–1.37 T Coercity 860–995 kA m Relative permeability 1.056–1.26 Conductivity 5882–9090.9 S m
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