Cases of coupled vibrations and prametric instability in rotating machines
Abstract: The principal task in this research project was to analyse the causes and consequences of coupled vibrations and parametric instability in hydropower rotors; where both horizontal and vertical machines are involved. Vibration is a well-known undesirable behavior of dynamical systems characterised by persistent periodic, quasi-periodic or chaotic motions. Vibrations generate noise and cause fatigue, which initiates cracks in mechanical structures. Motions coupling can in some cases augment the stability characteristics of a rotating machine, but it can also be a source of instability that causes self-excited vibrations. In this thesis, motions coupling due to a bearing’s design, gyroscopic effect and geometric misalignment in rotating components were studied. The performed studies include mathematical modelling and numerical simulation of the above named sources of motions coupling. Experiments were also performed in order to evaluate the derived analytical models. Plain cylindrical hydrodynamic journal bearings cross couple the rotor translational motions. This cross coupling is the main source of oil induced instability. The inherent nonlinearity of plain cylindrical hydrodynamic journal bearings becomes strong for eccentricities greater than 60% of the bearing clearance, where most existing linear models are not able to predict accurately the rotor trajectory. Therefore, the journal bearing impedance descriptions method, a method that is valid for all bearing eccentricities and aspect ratios, was used to analyse the rotor steady-state imbalance response. Strong nonlinearities together with cross coupling are the source of complex dynamics in fluid-film journal bearings. The simulation 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 speed at which oil induced instability occurs, also called the instability threshold speed, will depend upon the low stability characteristics of the less loaded bearing for an offset rigid rotor. However, for a flexible rotor the gyroscopic coupling effect will increase the instability threshold. The gyroscopic coupling effect not only increases the instability threshold, but the journal trajectories’ magnitude also significantly increases. This is normally not a preferable condition since high vibrations will induce heat and stress in babbited bearings. Adding rotor imbalance would enable the system to be operated beyond its threshold speed of instability with reduced vibration amplitudes. A tilting-pad combi-bearing is a bearing designed as a combination of both tilting-pad journal and thrust bearings. Thrust bearing is a component used in vertical rotating machines and shafts designed to transmit thrust, e.g. hydropower rotors and aircraft engines. The total axial load is normally carried by one thrust bearing. In hydropower applications, the influence of the combi-bearing is strongly simplified in the rotor dynamic modelling. The derived linear model shows that the combi-bearing couples the rotor’s lateral and angular motions at the contact point between the combi-bearing and the rotor. However, if the thrust bearing’s pads arrangement is not symmetrical or if all the pads are not angularly equidistant, the rotor vertical (axial) and angular motions are also coupled. This last case of coupling will also occur if the axial equivalent stiffness is not evenly distributed over the thrust bearing. A defected pad or unequal hydrodynamic pressure distribution on the pads’ surfaces may be the cause. The Porjus U9’s simulation results show that the combi-bearing influences the dynamic behavior of the machine. The rotor motions’ coupling due to combi-bearing changes the system’s natural frequencies and vibration modes. Introducing an angular misalignment in the combi-bearing’s rotating collar will generate an asymmetry in the rotor system at the combi-bearing’s location. The rotor system’s stiffness in its two translational directions differ at the combibearing’s location. Constant parameters and/or coefficients in rotating asymmetric structures appear to change with time when observed in the stationary frame. These time dependent parameters (coefficients) are the source of parametric instability in rotating systems. If the collar angular misalignment is located in one plane, all rotor motions in this plane at the contact point between the combi-bearing and the rotor will be coupled. A parametric instability is observed within certain ranges of the rotor speeds, depending on the magnitude of the angular misalignment. The studied cases of motions coupling due to plain cylindrical hydrodynamic journal bearings, motions coupling due gyroscopic effect, motions coupling due to tilting-pad combi-bearing and motions coupling due to manufacture or assembling errors in rotating components are cases that can be encountered in hydropower rotors. The results have revealed in particular that if the combi-bearing is manufactured or assembled with a certain angular misalignment, this may cause a parametric instability in the hydropower rotor. The parametric instability can even occur below the rotor critical speed, which would cause problems for undercritical machines as hydropower plants. The outcomes of these studies will contribute in further understanding of vibration problems and particularly in helping to improve and sustain the functionality of new and existing hydropower plants in Sweden.
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