Although the dynamics of vertical rotor bearing systems have been studied, the interaction between vertical rotors, bearings, and supporting structures - such as casings, bearing brackets, and foundations, remains less explored. This study presents a combined experimental and numerical investigation of a coupled vertical rotor system, incorporating a nonlinear, speed- and eccentricity-dependent bearing. The novelty lies in the description of a complex, vertical, rotor-bearing-support system incorporating a nonlinear journal bearing model, to capture the effects of the rotor’s vertical orientation, as typical of hydropower applications. The system features an elastic mid-span rotor supported by a flexible tower structure. The four-shoe tilting pad bearings impose significant stiffness variations and nonlinearities, connecting the stationary and rotating components. Modal analysis identifies the critical speeds of the flexible supporting structure, and simulations in the time domain are conducted for various run-up conditions, focusing on the bearing response across the structure’s first two natural frequencies. The results show qualitative and quantitative agreement between the experimental and simulated responses, highlighting the distinct dynamic behaviors of the upper and lower bearings. The bearing response at the structure’s first critical speed is studied and demonstrates improved accuracy during critical conditions. This model builds on established methods to accurately represent vertical rotor dynamics with nonlinear, eccentricity- and speed-dependent bearing models, while extending its applicability to more complex systems by incorporating bearing support flexibility, effectively providing a framework for simulating systems such as complete hydropower units.

In 2015, a 45 MW vertical hydropower machine exhibited excessive vibration after refurbishment. Measurements revealed a substantial bearing clearance at the lower generator guide bearing. Consequently, the bearing was unable to generate sufficient opposing force to drive the rotor toward the bearing center, resulting in more pronounced overall system vibration. Addressing this challenge required a cost-effective and feasible solution for mitigating the vibration problem. To this end, a design modification was implemented wherein the lower generator guide bearing (originally a sleeve bearing) was modified to a four-lobe bearing by offsetting the two halves of the bearing twice in two axes. Numerical simulations and experimentations were conducted, and the dynamics of the machine before and after the design modification were investigated. Both the simulation and experimental results showed that the machine with the four-lobe bearing improved the system stability and reduced the vibration amplitudes. The numerical simulation result demonstrated that, due to the design modification, the first and second critical speeds were effectively eliminated for a speed range of up to three times the nominal speed. Furthermore, for nominal operation with unbalanced magnetic pull, the four-lobe bearing provided a stability advantage in terms of the modal parameters relative to the original sleeve bearing.

This paper studies the dynamics of a 150 MW hydropower rotor system, in whichan elastic structure supports the generator guide bearing. A non-linear large-displacement force-deflection relationship for the bearing support is derived withinthe elastic range under quasi-static conditions, which is incorporated into theequations of motion for a rotor dynamic model. The proposed model includestwo tilting pad journal bearings, along with nonlinear magnetic interaction andunbalance forcing acting on the generator. The equation of motion is solved bytime-integrating procedures to study the system’s dynamics. A drastic softeningeffect is observed for the bracket within the elastic range, in which the nominalvalue represents merely 10% of the bearing stiffness. It is concluded that theexcessive support flexibility significantly reduces the lowest pair of frequencies,diminishes the damping ratios and alters the orbit characteristics during operation.Furthermore, the amplified eccentricity increases the risk of instability due to theradial magnetic pull.

Hydropower generators are normally considered rigid in rotordynamic studies; nevertheless, according to previous studies, they should be regarded as elastic, and this will affect the system's dynamics. The interaction between the magnetic flux of the stator and rotor creates attractive forces between the parts that would balance to zero at the centered position. However, asymmetries in the magnetic flux create an uneven distribution of forces that threaten the machine's life; these forces are non-linearly dependent on the gap between the rotation and stationary part of the generators, although it has been considered linear for simplification. This study focuses on how the dynamics of the rotor are affected by static eccentricity and unbalanced masses while considering nonlinear electromagnetic forces. The rotor has been modeled using curved beams and attached to the rotor-spider by connecting plates, allowing the expansion of the ring due to centrifugal and thermal loads. The electromagnetic forces are modeled as a sum of exponential functions. Bifurcation diagrams and Poincaré maps are employed to analyze the stability of the generator. 

Rotating machines may encounter rubbing due to contact between stationary and rotating structures, leading to large mechanical vibrations that can cause catastrophic failure. In hydropower machines, rubbing contact could occur for many reasons, such as mass imbalance or mechanical/electrical misalignments. This paper investigates whether squeeze film dampers can improve the contact dynamics of a 45 MW hydropower unit. The squeeze film dampers were installed in series with the upper and lower generator guide bearings, and the reaction forces were predicted based on short bearing approximation. A finite element model was established, and the equation of motion of the rotor–stator system was solved numerically using a MATLAB inbuilt function (ode23tb), taking the rotor speed, the retainer spring stiffness, and damper clearance as control parameters. The simulation results are presented using a bifurcation diagram, Poincaré map, orbit, frequency spectrum and maximum contact force. The results indicated that squeeze film dampers improved the damping characteristics of the hydropower machine and reduced the vibration amplitudes. Consequently, it curtailed the risk of rubbing contact at a critical speed for larger range of load cases, ensuring safe operations. Besides, retainer spring stiffness and damper clearance play an important role in the contact dynamics of the hydropower machine with squeeze film damper. 

Hydropower generators withstand multiple forces from diverse sources during operation. To ensure their stability and safe performance, numerical tools are developed to characterize their dynamic properties. Traditionally, generators are assumed to be rigid in rotordynamic analyses. However, the measurements in power stations challenge this assumption. This article proposes a novel approach to modeling hydropower generators with floating rotor rims using a three-dimensional (3-D) Finite Element Method, aiming to study their dynamic performance and properties, including natural frequencies, the modes of vibrations, and expansion due to centrifugal and electromagnetic forces, with the goal of improving the reliability of modern designs. Both this approach and employing a two-dimensional (2-D) model using curved beams result in similar in-plane natural frequencies and the expansion of the rotor rim due to centrifugal forces. However, the 3-D model can be used to calculate the out-of-plane natural frequencies and modes, to model the dynamics of complex geometries, and to perform stress evaluation and fatigue analysis.

For rotordynamic analysis of hydropower units, the generator is treated as a rotating rigid body. However, previous studies have confirmed that certain designs of generators are elastic, so the complex geometry of generators cannot be considered rigid. This work produced a model of hydropower generators with floating rotor rims, consisting of a rigid hub and a flexible rotor rim coupled with flexible connections. The model takes into account the influence of centrifugal and Coriolis effects, and the electromagnetic interaction between rotor and stator. The model also reproduces the dynamics of the generator with static and dynamic eccentricities. A generator prototype was employed to test the model, showing its different applications. Once validated by empirical data, this model could be used when designing generators.

Rotodynamic simulation of complex or/and large systems, for instance hydropower machines, may consist of models with many degrees of freedom and require multidisciplinary computations such as fluid-thermal-structure interactions or rotor-stator interactions due to electromagnetic forces. Simulating such systems is often computationally heavy and impractical, especially in the case of optimization or parametric study, where many iterations are required. This has, therefore, created a need for simplified dynamic models to improve computational efficiency without significantly affecting the accuracy of the simulation result. The purpose of this paper is to present simplified coordinate transformation matrices for journal bearings in vertical rotors, which require less computational effort. Matrix multiplications, which appear during coordinate transformation, were eliminated, and the bearing stiffness and damping matrices in the fixed reference frame were represented by local coefficients instead. The dynamic response of a vertical rotor with eight-shoe Tilting pad journal bearings was simulated using the proposed model for two operational conditions, i.e., when the rotor was spinning at constant and variable speeds. The results from the proposed model were compared to those from the original model and validated through experiments. The conclusion was that the presented simulation model is time efficient and can effectively be used in rotordynamic simulations and analyses.

Dynamic models for hydropower generators treat the rotor part as a rigid body; however, many studies have shown the opposite. The electromagnetic force distribution of deformed rotors is uneven, creating Unbalance Magnetic Pull, causing high forces on generator components leading to a risk of fatigue, therefore shortening the life of machines. Unbalancing masses can worsen the asymmetries of the rotor, which would further increase the effect of the electromagnetic interactions. This paper evaluates the rotor response using different unbalancing masses at the rotor and at the poles to quantify their impact in displacements and exciting frequencies. The model employed in this paper is based on the equation of motion derived using Lagrange equations in both co-rotating and stationary frames of reference, considering the effects of Centrifugal loads, Coriolis, and magnetization of poles. Different unbalancing mass placements affect different variables; extra weights in the poles contribute predominantly to the deformation of the rim, while the unbalance in the shaft affects the position of the shaft; a combination of placements was also studied. The simulations were performed and compared with and without radial electromagnetic forces, showing how the presence of magnetized poles further deforms the shapes of the rotor.