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.

Purpose: In this paper, the aim is to present a modeling strategy for a flexible rotor/active magnetic bearing (AMB) system with non‐collocation. Special attention is paid to the vibration reduction and the stable passage through the first critical speed.Design/methodology/approach: The finite element method based on Euller‐Bernoulli beam theory is applied in the formulation of the rotor model. Since rotor/AMB systems are complex mechatronic systems, reduced order approach is used in the control system design. This study applies the modal decomposition method and the modal truncation method, thus retaining the lower order bending modes. The obtained numerical results are compared with the measured open loop frequency responses and the existing differences are compensated in order to obtain accurate numerical model.Findings: Frequency response of the entire system model (flexible shaft, actuators, power amplifiers and sensors) with amplitudes expressed in rotor lateral displacements can be verified by the measured frequency responses. The deviations in the amplitude and phase diagrams are then successfully corrected using the appropriate model modifications.Practical implications: The results of this research find direct applications in flexible rotors supported by AMBs, e.g. high speed spindles, turbo molecular pumps, flywheel energy storage systems, etc. The presented procedure can be especially valuable in the design of model based controllers.Originality/value: An AMB system model is developed and presented in this paper, in conjunction with a systematic description of an efficient procedure for the elimination of the typical mismatches between the simulation and experiment. Firstly, rotor/AMB test rig is stabilized with an appropriately tuned PID controller and an open loop frequency response is obtained for such a system. This response is then compared to corresponding simulation results for which mismatches are identified and eliminated thus yielding an accurate model of the system.

In this paper, the calculation of the dynamic response to harmonic transverse excitation of cantilever Euler-Bernoulli beam carrying a point mass with the mode superposition method is presented. This method uses mode shapes and modal coordinate functions which are calculated by separation of variables and Laplace transformations. The accuracy of defined expressions is confirmed on examples with and without Rayleigh damping. All calculations are in good agreement with the results of commercial software based on finite element methods.

The International Maritime Organization (IMO) mandates a reduction in carbon dioxide emissions from 2008 levels by at least 40% by 2030, prompting the widespread adoption of slow steaming and engine de-rating strategies. This study investigates the fatigue life of marine propulsion shafts under slow steaming conditions, focusing on the interplay between torsional and bending vibrations. A finite element (FE) model of a low-speed two-stroke propulsion system is developed, incorporating torsional and lateral excitation sources from both the engine and propeller. Vibrational stresses are computed for multiple operating conditions, and fatigue life is assessed using both the conventional Det Norske Veritas (DNV) methodology and a proposed biaxial stress approach. Results indicate that while torsional vibrations remain the primary fatigue driver, bending-induced stresses contribute marginally to the overall fatigue life. The proposed methodology refines high-cycle fatigue (HCF) assessment by incorporating a corrected S-N curve and equivalent von Mises stress criteria. Comparisons with classification society standards demonstrate that existing guidelines remain valid for most cases, though further studies on extreme alignment deviations and dynamic bending effects are recommended. This study enhances understanding of fatigue mechanisms in marine shafting and proposes a refined methodology for improved fatigue life prediction.