The active magnetic bearing(AMB) suspends the rotating shaft and maintains it in levitated position by applying controlled electromagnetic forces on the rotor in radial and axial directions. Although the development of various control methods is rapid, PID control strategy is still the most widely used control strategy in many applications, including AMBs. In order to tune PID controller, a particle swarm optimization(PSO) method is applied. Therefore, a comparative analysis of particle swarm optimization(PSO) algorithms is carried out, where two PSO algorithms, namely (1) PSO with linearly decreasing inertia weight(LDW-PSO), and (2) PSO algorithm with constriction factor approach(CFA-PSO), are independently tested for different PID structures. The computer simulations are carried out with the aim of minimizing the objective function defined as the integral of time multiplied by the absolute value of error(ITAE). In order to validate the performance of the analyzed PSO algorithms, one-axis and two-axis radial rotor/active magnetic bearing systems are examined. The results show that PSO algorithms are effective and easily implemented methods, providing stable convergence and good computational efficiency of different PID structures for the rotor/AMB systems. Moreover, the PSO algorithms prove to be easily used for controller tuning in case of both SISO and MIMO system, which consider the system delay and the interference among the horizontal and vertical rotor axes.

This paper presents a comparison between two analytic methods for the determination of natural frequencies and mode shapes of Euler-Bernoulli beams. The subject under scrutiny is the problem of a beam supported by an arbitrary number of translational springs of varying stiffness, which is solved first by the method of Laplace transform and then by the Green’s function method. The two methods are compared on the level of algebraic equivalence of the resulting formulas and these are compared to the results obtained by FEM analysis for the case when the beam is supported with a single spring. It can be shown that both methods result in equivalent algebraic expressions, whose results are to be regarded as accurate for any given set of boundary conditions. This is also verified by means of FEM analysis, whose solutions converged to almost identical values. Hence, the two methods are found to be equally accurate for the calculation of natural frequencies and natural modes. This paper also brings a new formulation for the mode shape equation, obtained by the Laplace transform method, which to the best of the authors’ knowledge has not been reported in existing literature. Additional comments about the advantages and disadvantages of the two employed analytical methods are given from the standpoint of mathematical structure and practicality of implementation, which is also supplemented by a comment on the comparative advantages and disadvantages of FEM analysis.