34 M. V. F. de Oliveira et al. Fig. 4.2 Main components of an AMB system Fig. 4.3 Rotor finite element model node, two displacements and two rotations along the x and y directions. Figure 4.3 presents the rotor model, where the axial positions of the sensors and AMB’s are also indicated. Due to the size of this model, the pseudo modal method [18] was used to reduce the system, thus keeping only the first 10 rotor mode shapes, which were identified as relevant for the dynamic behavior of the system as pointed out by Hankel singular value analysis. According to [20], the dynamic behavior of power amplifiers can be accurately modeled by a first order transfer function, as given by Eq. (4.1): Gamp DKamp 1 1 !camp s C1 (4.1) where Gamp is the power amplifier transfer function, Kamp is the system gain and!camp is the amplifier frequency bandwidth. According to the manufacturer: Kamp Š1.0A/V; !camp Š1000 Hz. We are interested on the the frequency corresponding to an amplitude reduction of 3 dB (frequency bandwidth) (Fig. 4.4). The time delay of the AMB system is mostly due to the process of conversions A/D and D/A. In the literature [13, 20], this phenomenon is usually modeled by a second order Padé approximation, as presented in Eq. (4.2). This approach considers that the time delay is given by 1.5Ts,where Ts is the inverse of the acquisition rate. In this case, Ts D1/10000 s andnD20. e Tss 1 Tss 2 n n = 1C Tss 2 n n (4.2)
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