# Coupled vibrations in horizontal and vertical rotor-bearings systems

Sammanfattning:

For dynamical systems having several degrees of freedom, motion in one direction can induce motion in the other and/or vice versa. This means that there is a certain coupling between these two motions. Coupling can in some cases be a source of instability that causes self-excited vibrations in rotating machinery. In modeling hydropower rotors, couplings other than those that are the result of gyroscopic effect are normally not considered. This is due to the complexity of the reasons for coupling which mainly depends on machinery hardware, for example, the bearing's design (type) and the asymmetry in machine components. In this thesis, couplings due to bearings and gyroscopic effect were studied analytically and numerically. The performed studies include mathematical modeling and numerical simulation of some cases examples. Plain cylindrical hydrodynamic journal-bearing was modeled as a fluid-film lubrication separating the rotor from the stationary rigid bearing. Both nonlinear and linear fluid-film forces were considered in the analyses. In case of tilting-pad bearings, the fluid-film and the flexible support structures were modeled as linear stiffness while neglecting the pads inertia and the fluid-film damping.

Plain cylindrical hydrodynamic journal-bearings provide high damping to the rotor system, but they also cross couple the rotor translational motions. Cross coupling is the main source of oil induced instability; therefore, the rotor speed should not exceed the speed at which oil-induced instability occurs. The inherent nonlinearity of plain cylindrical hydrodynamic journalbearings becomes strong for eccentricities greater than 60% of the bearing clearance, where most existing linear models are not able to accurately predict the rotor trajectory. Strong nonlinearities together with cross coupling are the source of complex dynamics in fluid-film journal bearings. Paper A concerns analysis of the dynamic behavior of a rigid symmetric rotor that is supported by two identical finite-length journal bearings at high eccentricities. The journal bearing impedance descriptions method, a method that is valid for all bearing aspect ratios and all eccentricities, was used to evaluate linear analysis of the rotor steady-state imbalance response. The results show that linear bearing models derived from the nonlinear impedance descriptions of the Moes-cavitated (π - film ) finite-length bearing can predict the steady-state imbalance response of a rigid symmetric rotor that is supported by two identical journal-bearings at high eccentricities. This is, however, only the case when operating conditions are below the threshold speed of instability and when the system has period one solutions. The error increases in the vicinity of resonance speed.

The gyroscopic coupling effect on oil whirl instability and journal trajectories was analyzed in Paper B. The same linear and nonlinear bearing models from Paper A were reused here, and the flexible non-symmetric rotor that is supported by two identical finite-length journal-bearings was modelled by finite element method. The results show that the instability threshold of a rigid non-symmetric rotor-bearing system depends on the low stability characteristics of the less loaded bearing. However, when shaft flexibility and the gyroscopic coupling effect are taken into account, the instability threshold increases. The gyroscopic coupling effect not only increases the instability threshold, the magnitude of journal trajectories also significantly increases. This is normally not a preferable condition since high vibrations will induce heat and stress in babbited bearings. Rotor imbalance has a positive effect on flexible non-symmetric rotors; it enables the rotor system to be operated beyond its threshold speed of instability with reduced vibration amplitudes.

Tilting pad journal-bearing has low cross coupling between the rotor's translational (lateral) motions. Vertical rotors and shafts designed to transmit thrust are equipped with thrust bearing. Paper C focuses particularly on modelling and analyzing the thrust bearing's dynamic influence on vertical rotors. A case study on an existing vertical hydroturbogenerator is presented. Results show that the tilting-pad thrust bearing influences the system's second and third natural frequencies. But in this case, the system's first natural frequencies were not influenced by the thrust bearing. The rotor second natural lateral vibration mode shows that the rotor system without thrust bearing has larger vibration amplitude at the exciter location than the rotor system with thrust bearing. The thrust bearing stiffening effect at rotor second natural bending mode has also resulted in reduced vibration amplitude of the rotor at the exciter location. The main observation is that the thrust bearing generates a stiffening moment which is directly proportional to the rotor's angular motions at thrust bearing location.

The results obtained from the conducted studies are useful during the design process for new hydropower rotor-bearing systems and for maintaining old existing hydropower plants. The developed models can serve as a simulation tool during design modifications or during analysis of failures. Due to the large scale of real hydropower units, simulations are useful because they are more time and cost-efficient than running full-scale experiments. They also facilitate analysis of a large number of operating conditions and design modifications.

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