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Multiple Solutions in an Amplitude Limited Jeffcott Rotor Including Rubbing and Stick-Slip Effect
Abstract
The non-linear behaviour of rub-impact rotors have been studied in several papers. In such systems rich dynamics have been found together with the coexistence of solutions within some specific parameter ranges. In this paper an attempt is made to find all stable solutions for an amplitude limited Jeffcott rotor including rubbing and stick-slip effect. The recently suggested “multi bifurcation diagram method” is used to find and extract stable sets of bifurcation diagrams. A system is chosen where the linear stationary amplitude only exceeds the clearance in a narrow region near the natural frequency. Therefore large regions in frequency are expected to have only the linear stationary response. The results show that it is only for very low frequencies that one single solution exists. Even though periodic motions are dominant, there exist large ranges in frequency with quasi-periodic or chaotic motions. For the studied cases, three coexisting stable solutions are most common. In one case as many as four stable solutions was found to coexist. For rotors with large clearances (no impacts necessary) it is still possible to find several coexisting motions. For all cases the stick motion is the most severe one with large amplitudes and high backward whirl frequencies. In real situations the consequence of this stick motion is machine failure. These high amplitude motions were found to be stable over large frequency ranges. From the stability analysis it was found that this rolling motion can be avoided by low spin speed, low contact stiffness, low coefficient of friction, small ratio of disc radius/clearance or high damping ratio. In a design situation the parameters are seldom known with high accuracy. Therefore, it is of interest to know all solutions for parameter intervals. The multi-bifurcation diagram can be used in such situations to design a robust machine or at least be prepared for unwanted dynamics.
... When all solutions are known, the designer can hopefully find parameter ranges with acceptable solutions or at least know what kind of problems to expect. The method has been developed in the papers by Aidanpää [6][7][8] but was not described in detail. ...
In mechanical applications it is essential that unexpected dynamics are avoided. The industry wants to build reliable machines that are not sensitive to initial conditions. Therefore, a simple method has been developed to extract all sets of stable bifurcation diagrams. The method gives a designer a good overview of possible dynamics and thereby the possibility to select a safe operating region. The method is described and demonstrated with a rub-impact rotor. The practical usage of this method is to help the designer to determine if parameter ranges exist where coexistent solutions will appear. Thereby one can design the system to work in parameter ranges where only one acceptable solution exists. .
Damages due to contact between the runner and the discharge ring have been observed in several hydroelectric power units. The damage can cause high repair costs of the runner and the discharge ring as well as considerable production losses.In this paper a rotor model of a 45 MW hydropower unit is used for the analysis of the rotor dynamical phenomena occurring due to contact between the runner and the discharge ring for different grades of lateral force on the turbine and bearing damping. The rotor model consists of a generator rotor and a turbine, which is connected to an elastic shaft supported by three isotropic bearings. The discrete representation of rotor model consists of 32 degrees of freedom; to increase the speed of the analysis, the size of the model was reduced with the IRS method to a system with 8 degrees of freedom.Results are presented in bifurcation diagrams, maximum contact force, Poincaré map and phase portrait. Simulations indicate that the contact forces between the turbine and the discharge ring are large, with considerable risks for serious damage as a consequence. The analysis shows that the risk for contact and damage are large for relatively small lateral turbine loads when the gap between the turbine and the discharge ring is small and the contact stiffness is high.
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