Miguel Moreira

Nonlinear analysis of the orbital motions of immersed rotors using a spectral/Galerkin approach

Abstract We recently developed a symbolic-numerical formulation for the nonlinear planar motion of rotors under fluid confinement, based on a spectral/Galerkin approach, for gap geometries of about δ = H / R ≈0.1––where H is the average annular gap and R is the rotor radius. Results showed a quite good agreement between the class of approximate models generated, the corresponding analytical exact planar model and experiments. This methodology can be almost entirely automated on a symbolic computing environment. In the present paper this symbolic-numerical spectral/Galerkin procedure is extended in order to deal with nonlinear orbital motions–– X ( t ) and Y ( t ) taking place in orthogonal directions. Numerical simulations performed over a centered rotor configuration maintained by nonisotropic supports ( K st Y / K st X =0.4, where K st X and K st Y stand for the structural stiffnesses), which exhibit interesting dynamics, show a quite good agreement between this type of approximate models and the corresponding analytical exact (but quite involved) model, developed in the past by the authors. With the proposed symbolic-numerical approach one can obtain accurate nonlinear dynamical formulations enabling the study, understanding and prediction of nonlinear orbital rotor dynamics.

Experimental Validation of Theoretical Models for the Linear and Nonlinear Vibrations of Immersed Rotors

Vibration of rotating shafts has been studied for different gap geometries, ranging from bearing configurations to pump systems. This paper deals with the rotor-flow dynamics of immersed shafts under moderate confinement-clearance gap about S=#=O.l( h w ere H is the average gap and R is the rotor radius). Following simplified assumptions, analytical models for the linearized forces, for both centered and eccentric immersed rotors have been developed as well as a theoretical nonlinear model which fully describes the nonlinear flow terms. These models were supported by encouraging results from preliminary experiments. In the present paper, we discuss some recent and representative results of an extensive series of tests performed on a small-scale model, in order to assert the validity of our theoretical models. From the overall experimental programma, the following conclusions emerged: (1) The linearized bulk-flow model is adequate, provided the dissipative effects are duly accounted for using an empirical friction coefficient to empirically model the turbulent stresses. Such predictions are quite accurate if the system is working at low rotor eccentricities and far from the instability boundaries. (2) However, for large rotor eccentricities and for dynamic regimes near the linear instability, the fully nonlinear model ‘Visiting researcher at Institute of Nuclear Technology. leads to better predictions. Obviously, these effects are instrumental to obtaining reasonable predictions for all post-stable motion regimes. (3) When discrepancies arise, the nonlinear model was usually found to be conservative. INTRODUCTION The effects of co-rotating annular flows on rotor vibrations have been studied extensively by many researchers, mostly in connection with bearings and seals [see for instance (Chills, 1993)]. However, flow structure interaction can also lead to significant effects in less common devices. In this paper we will address the dynamical behavior of immersed rotors, such as found in fast-breeder nuclear reactor pumps (for circulating the liquid sodium), and other ap plications. In such components, the clearance ratio 6 = H/R (where H is the average gap and R is the rotor radius) is typically about 0.1 -one or two orders of magnitude higher than the clearances found in bearings and seals. As a consequence, the flow is quite turbulent, inertial effects are then of prime importance and cannot be neglected as assumed in the basic Reynolds equation approach. Also, the shaft length subject to fluid forces is quite significant, these combined effects leading to specific rotordynamic properties. Using simplified assumptions for the flow, we have developed analytical models for the linearized forces, for both centered and eccentric rotors [ (Axisa and Antunes, 1992), (Antunes et al., 1996) and (Moreira et al., 2OOO)]. Because the flow was modeled as being two-dimensional, it was possible to extend our analytical solutions to fully account for the nonlin-

An improved linear model for rotors subject to dissipative annular flows

Abstract In a previous paper, Antunes, Axisa, and co-workers, developed a linearized model for the dynamic of rotors under moderate fluid confinement, based on classical perturbation analysis, covering two different cases: (i) dissipative motions of a centered rotor; (ii) motions of an eccentric rotor for frictionless flow. Following the same procedures and assumptions, we derive here an improved model for the more general case of dissipative linearized motions of an eccentric rotor . Besides the rotor motion variables, a new variable—which can be interpreted as the fluctuating term of the average tangential velocity—is introduced, yielding an additional eigenvalue in the linear analysis. The new variable introduced, which is coupled with the rotor motions, is very convenient when frictional effects are not neglected. Under dissipative flows, a richer modal behavior is highlighted, which can be related to delay effects of the flow responses to the rotor motions. Our approach can be applied as well to other flow-excited systems, for example, those subjected to axial or leakage flows. Because rotor-dynamics are strongly dependent on the mean rotor eccentricity, the adequacy of this (or any other) model rely on using the actual value for such parameter.

Fluid-Coupled Vibrations of Immersed Spent Nuclear Racks: A Nonlinear Model Accounting for Squeeze-Film and Dissipative Phenomena

Fluid-coupling effects lead to a complex dynamical behavior of immersed spent fuel assembly storage racks. Predicting their responses under strong earthquakes is of prime importance for the safety of nuclear plant facilities. In the near-past we introduced a simplified linearized model for the vibrations of such systems, in which gap-averaged velocity and pressure fields were described analytically in terms of a single space-coordinate for each fluid inter-rack channel. Using such approach it was possible to generate and assemble a complete set of differential-algebraic equations describing the multi-rack fluid coupled system dynamics. Because of the linearization assumptions, we achieved computation of the flow-structure coupled modes, but also time-domain simulations of the system responses. However, nonlinear squeeze-film and dissipative flow effects, connected with very large amplitude responses and/or relatively small water gaps, cannot be properly accounted unless the linearization assumption is relaxed. Such is the aim of the present paper. Here, using a similar approach, we generalize our theoretical model to deal with nonlinear flow effects. Besides that the proposed methodology can be automatically implemented in a symbolic computational environment, it is much less computer-intensive than finite element formulations. Using the proposed technique, computations of basic flow-coupled rack configurations subjected to impulse excitations are presented, in order to highlight the essential features of such systems as well as the relevance of squeeze-film and dissipative effects. Finally, more realistic simulations of complex system responses to strong seismic excitations are presented and discussed.Copyright

Nonlinear Analysis of the Orbital Motions of Immersed Rotors Using a Spectral/Galerkin Approach

We recently developed a symbolic-numerical formulation for the nonlinear planar motion of rotors under fluid confinement, based on a spectral/Galerkin approach, for gap geometries of about δ = H/R ≈ 0.1 — where H is the average annular gap and R is the rotor radius. Results showed a quite good agreement between the class of approximate models generated, the corresponding analytical exact planar model and experiments. This methodology can be almost entirely automated on a symbolic computing environment. In the present paper this symbolic-numerical spectral/Galerkin procedure is extended in order to deal with nonlinear orbital motions — X (t) and Y (t) taking place in orthogonal directions. Numerical simulations performed over a centered rotor configuration maintained by non-isotropic supports (Ky st /Kx st = 0.4, where kx st and ky st stand for the structural stiffnesses), which exhibit an interesting dynamics, show a quite good agreement between this type of approximate models and the corresponding analytical exact (but quite involved) model, developed in the past by the authors. With the proposed symbolic-numerical approach one can obtain accurate nonlinear dynamical formulations enabling the study, understanding and prediction of nonlinear orbital rotor-dynamics.Copyright

A Theoretical Series Solution for the Dynamics of Finite-Length Bearings and Squeeze-Film Dampers

In this paper we develop a non-linear dynamical solution for finite length bearings and squeeze-film dampers based on a Spectral-Galerkin method. In this approach the gap-averaged pressure is approximated, in the lubrication Reynolds equation, by a truncated double Fourier series. The Galerkin method, applied over the residuals so obtained, generate a set of simultaneous algebraic equations for the time-dependent coefficients of the double Fourier series for the pressure. In order to assert the validity of our 2D–Spectral-Galerkin solution we present some preliminary comparative numerical simulations, which display satisfactory results up to eccentricities of about 0.9 of the reduced fluid gap H/R. The so-called long and short-bearing dynamical solutions of the Reynolds equation, reformulated in Cartesian coordinates, are also presented and compared with the corresponding classic solutions found on literature.© 2003 ASME