Nonlinear Response and Bifurcations Analysis of Rotor-Fluid Film Bearing Systems

Nonlinear response, bifurcations and stability of rotor-fluid film bearing systems are studied using various numerical investigation schemes such as autonomous/non-autonomous shooting, arc-length continuation, direct numerical integrations, Poincaré sections, Lyapunov exponents, etc. Two types of hydrodynamic bearings, a floating ring bearing (FRB) and a tilting pad journal bearing (TPJB), are employed in this study. The nonlinear characteristic of each bearing is analyzed as supports of a rigid rotor system as well as a flexible rotor system. Depending on the existence of the unbalance force on the rotor/disks, autonomous (free vibration) and non-autonomous responses (mass unbalanced excitation) are both identified, and the nonlinear reaction force produced on the lubricant layer is obtained using the finite element method. In addition to isoviscosity lubricants, thermo-hydrodynamic lubricant model is developed to investigate thermal effects on rotordynamic bifurcations; in the procedure, a variable viscosity Reynolds equation and the energy equation are solved simultaneously. For computation efficiency in the analytical bifurcation study, an advanced shooting algorithm, which is combined with the deflation theory and the parallel computing strategy, is proposed for both the autonomous and the non-autonomous cases. In the study with flexible rotors, the finite element based beam models are employed and the model reduction technique such as Component Mode Synthesis is utilized to condense the system degree of freedom. This dissertation consists of four main discussions regarding: 1) nonlinear response and bifurcations of a rigid rotor supported by FRBs; 2) effects of a thermo-hydrodynamic (THD) FRB model on rotordynamic bifurcations; 3) nonlinear response and bifurcations of a rigid rotor supported by TPJBs; 4) extension of study to general, complex, multi-mass rotor beam models. In case 1), multiple coexistent solutions and bifurcation scenarios are identified, and those are depended on the ratio of floating ring length to diameter (L/D). Numerical illustrations regarding jumps between two stable limit cycles and quenching large vibrations are demonstrated, and chaos is investigated with the aid of Lyanpunov exponent. In case 2), the Hopf bifurcation onset is strongly dependent on thermal conditions, and the saddle node bifurcation points are significantly shifted compared to the isothermal model. In addition, the unbalanced responses stability and bifurcation onsets are highly reliant on the lubricant supply temperature. In case 3), loci of bifurcations are identified, and heavily loaded bearings and/or high unbalance force may induce consecutive transference of response in forms of synchronous to sub-synchronous, quasi-periodic responses and chaotic motions. The periodic doubling bifurcations, saddle node bifurcations and corresponding local stability are reliably determined by selections of pad preload, pivot offset, and lubricant viscosity sets. In case 4), two industrial applications such as a turbocharger supported by FRBs and an eight-stage centrifugal compressor supported by TPJBs are numerically analyzed. The turbocharger shows that torus appears with Neimark-Sacker bifurcation events and the motions are dominant in the high speed ranges (>60,000rpm). In the compressor, sub-/super-synchronous motions are identified other than the ×1 synchronous response, and the appearance of each harmonic is highly depended on the selection of pad preload and pivot offset

Shooting/continuation based bifurcation analysis of large order nonlinear rotordynamic systems

This study introduces an improved numerical algorithm that is capable of analyzing nonlinear vibrations and bifurcations of general, finite, large-order rotordynamic systems supported on nonlinear bearings. An industrial rotor generally consists of several sections and stages, but numerical shooting/continuation method has been applied to a simple Jeffcott type rotor instead of complex models due to the computational burden of the numerical procedure; it becomes significant when the rotor combined with nonlinear finite bearing models. Here, some mathematical/computational techniques such as a deflation algorithm and the parallel computing are suggested for acceleration along with the conventional treatment of model reduction scheme. An eight-stage compressor rotor supported by two identical five-pad tilting pad journal bearings (TPJB) is selected as a mechanical model to test the numerical incorporation of the algorithms. The rotor beam is modelled with 35 nodes, 140 DOF based on Euler beam theory, and the fluid reaction forces from the two TPJB are calculated using simplex, triangular type finite meshes on the pads. In the numerical procedure, the shooting/continuation combined with the acceleration schemes identifies the solution curves of periodic responses and determines their stability. The orbital motions of coexistent responses are obtained from the solution manifolds

Shooting/continuation based bifurcation analysis of large order nonlinear rotordynamic systems

This study introduces an improved numerical algorithm that is capable of analyzing nonlinear vibrations and bifurcations of general, finite, large-order rotordynamic systems supported on nonlinear bearings. An industrial rotor generally consists of several sections and stages, but numerical shooting/continuation method has been applied to a simple Jeffcott type rotor instead of complex models due to the computational burden of the numerical procedure; it becomes significant when the rotor combined with nonlinear finite bearing models. Here, some mathematical/computational techniques such as a deflation algorithm and the parallel computing are suggested for acceleration along with the conventional treatment of model reduction scheme. An eight-stage compressor rotor supported by two identical five-pad tilting pad journal bearings (TPJB) is selected as a mechanical model to test the numerical incorporation of the algorithms. The rotor beam is modelled with 35 nodes, 140 DOF based on Euler beam theory, and the fluid reaction forces from the two TPJB are calculated using simplex, triangular type finite meshes on the pads. In the numerical procedure, the shooting/continuation combined with the acceleration schemes identifies the solution curves of periodic responses and determines their stability. The orbital motions of coexistent responses are obtained from the solution manifolds

Effects of Tilting Pad Journal Bearing Design Parameters on the Pad-Pivot Friction and Nonlinear Rotordynamic Bifurcations

This study numerically analyzes and investigates the effects of the bearing design parameters of a tilting pad journal bearing (TPJB) on the pad-pivot friction-induced nonlinear rotordynamic phenomena and bifurcations. The bearing parameters were set to the pad preload, pivot offset, spherical pivot radius, and bearing length to diameter (L/D) ratio. The Stribeck curve model (SCM) model was applied at the contact surface between the pad and the pivot, which varied to the boundary-mixed-fluid friction state depending on the friction condition. The rotor-bearing model was set up with a symmetrical five-pad TPJB system supporting a Jeffcott type rigid rotor. The fluid repelling force generated in the oil film between each pad and the shaft was calculated using a finite element method. The simulation recurrently conducted the transient numerical integration to obtain the Poincaré maps and phase states of the journal and pad with various bearing design variables, then the nonlinear properties of each condition were analyzed by expressing the bifurcation diagrams. As a result, the original findings of this study are: (1) The pad preload and pivot offset significantly influenced the emergence of Hopf bifurcations and the associated limit cycles. In contrast, (2) the pivot radius and L/D ratio contributed relatively less to the friction-induced instability. Resultantly, (3) all the effects diminished when the rotor operated under the larger mass eccentricity of the disc