Research On Bending-torsion Coupling Vibration And Bifurcation Analysis Of Two State Variables

As the developments of social economy, the requirement of electric isincreasing, which promotes the turbo-generator unit to large scale and large capacity.As the result, the vibration of the turbo-generator is more and more serious. Inactual system, for the existence of unbalanced mass and exciting force, thebending-torsion coupling vibration causes more and more attention. The research onthe bending-torsion coupling vibration of turbo-generator has some challenge.Further study on it will reveal the dynamical characteristics of the rotor system,increase the accuracy of fault diagnosis, and provide theoretical basis for parametricdesign.In this paper, the primary resonance of the rotor system with double disks andunbalanced mass is studied. The bending-torsional coupling motion equations of lowpressure cylinder-engine system are derived by using Lagrange equation, where thenonlinear factor is caused by unbalanced mass. By using average method, theanalytical solutions and bifurcation equations are obtained. And the differentbifurcation modals in persistent regions are obtained by singularity theory. After that,the motion equations are simulated by Runge-Kutta method. It is found that theunbalanced mass can arouse bending-torsion coupling vibration and the balancemore large the vibration more serious. Therefore, decreasing the unbalanced masscan reduce the bending-torsional coupling vibration. For the system, primaryresonance may be considered as the input energy of the torsional vibration, whichcan intensify the bending vibration. Therefore, for the actual system,bending-torsion coupling vibration should be considered.In subsequence, motion equations of low pressure cylinder-engine system underprimary resonance and combined resonance (the frequency of exciting force is thesum of natural frequency of bending vibration and the one of torsinal vibration) intosional direction are constructed, where anisotropic stiffness of shaft is considered.Similarly, the analytical solutions and bifurcation equations are obtained by usingaverage method. And the different bifurcation modals in persistent regions areobtained by singularity theory. The analytical solutions are tested by numericalcalculation. It is found that under the primary resonance and combined resonancethe amplitudes of bending vibration and torsional vibration both increase largely. Inthis case, primary resonance can be considered as the input energy of the torsionalvibration, and combined resonance can be considered as energy switch path. Whenthe primary resonance causes the torsional vibration of shaft, the energy istransmitted to bending vibration through switch path. Then the amplitude of bending vibration increases. For this case, the system can be destroyed both in bendingdirection and torsional direction. Therefore during the dynamical analysis of systemthe bending-torsion coupling vibration should be considered, and in the design phaseof the system this case is best to avoid.The bending-torsion coupling vibration of shaft caused by parallel misalignmentof flange coupling is analyzed, where the nonlinear factor is caused by unbalancedmass. The bending-torsion coupling motion equations are constructed for the case ofcombined resonance, i.e. the ratio of the frequency of exciting force, naturalfrequency of bending vibration and natural frequency of torsinal vibration is1:1:2.By using average method and singularity theory, the analytical solutions andbifurcation equations are obtained. And the numerical solutions are simulated byRunge-Kutta method. For the combined resonance (the ratio of the frequency ofexciting force, natural frequency of bending vibration and natural frequency oftorsinal vibration is1:1:2), there is not only input energy (the ratio of the frequencyof exciting force and natural frequency of bending vibration is1:1), but also energyswitch path (the ratio of natural frequency of bending vibration and naturalfrequency of torsinal vibration is1:2), which can cause the damage of the system.Therefore, in this case the bending-torsion vibration should be considered. Duringthe design of the system, the combined vibration is best to avoid.For the actual system, the state variables are usually constrained. There aremany studies on the one state variable singularity theory with constraints, whilelittle study on the two state variables singularity theory with constraints. In thispaper, the two state variables system with constraints is studied, where sevendifferent constraints are considered. The transition sets are obtained. In theunconstrained region, the bifurcation set and double limit set are confirmed with theones without constraints, but hysteresis set is different. The constraints arouse somenew transition sets. These new transition sets are associated with boundary.Therefore, they are called boundary induced transition sets. As an example, thebifurcation of a two dimensional system with constraints is analyzed. It is found thatthere are some new transition sets are induced by the constraints, which is a test ofthe bifurcation theory of two state variables system with constraints.