A 3D finite element model for the vibration analysis of asymmetric rotating machines

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Abstract

This paper suggests a 3D finite element method based on the modal theory in order to analyse linear periodically time-varying systems. Presentation of the method is given through the particular case of asymmetric rotating machines. First, Hill governing equations of asymmetric rotating oscillators with two degrees of freedom are investigated. These differential equations with periodic coefficients are solved with classic Floquet theory leading to parametric quasimodes. These mathematical entities are found to have the same fundamental properties as classic eigenmodes, but contain several harmonics possibly responsible for parametric instabilities. Extension to the vibration analysis (stability, frequency spectrum) of asymmetric rotating machines with multiple degrees of freedom is achieved with a fully 3D finite element model including stator and rotor coupling. Due to Hill expansion, the usual degrees of freedom are duplicated and associated with the relevant harmonic of the Floquet solutions in the frequency domain. Parametric quasimodes as well as steady-state response of the whole system are ingeniously computed with a component-mode synthesis method. Finally, experimental investigations are performed on a test rig composed of an asymmetric rotor running on nonisotropic supports. Numerical and experimental results are compared to highlight the potential of the numerical method.

Introduction

Understanding physics of rotating machines is essential in many application fields. For example, safety and service life in power plants rotating machinery such as turbo-alternator [1] or GT-MHR reactor project [2] are directly linked to a high-quality control of their vibratory behaviour. Basic rotordynamics modelling for design and predictive maintenance [3] is usually sufficient to solve most of the industrial problems encountered but there is still advanced topics to investigate [4]. One of them concerns the 3D global modelling of asymmetric rotating systems.

Behaviour of unsymmetrical large rotating machines includes here a lot of cases of practical importance such as dynamics of rotors with shape imperfections [5], vibratory signature of cracked rotors for on-line monitoring [6], [1], lateral instabilities of drill-strings [7]. Under the assumption of constant rotating speed, all these problems have in common to be linear periodically time-varying systems [4], i.e. governed by differential equations with periodic coefficients. Solutions of these equations are generally determined thanks to the well-known Floquet theory [8]. These solutions can be computed in the time domain by direct integration of the Floquet transition matrix as in [9]. An other possibility is to determine these solutions in the frequency domain by analysing the set of equivalent linear time-invariant equations obtained by the Hill expansion [10].

First modelling of unsymmetrical rotating machines were performed using simple rotating oscillators with anisotropic stiffnesses [5]. Indeed, for these academic models, Hill governing equations are easily established and solved [11]. Extension to the finite element models in stationary coordinates is referred in [12] for unsymmetrical rotors and in [4], [13] for both anisotropic rotors and supports. Still in the inertial frame, a modal analysis for periodically time-varying rotors is suggested in [6], [14] and allows reasonable computation time for simple rotor models (with shape imperfection, transverse cracks). However, the standard representation in the inertial frame limits our investigations to rotating shafts modelling with beam elements [2]. A complete 3D modelling of rotors described in the rotating frame [4] enables to capture a richer kinematics (blades, flexible discs, original shapes, etc.) and facilitates the dynamic studies of asymmetric rotors [15].

This paper focuses on the global vibratory behaviour of asymmetric rotating machines taking into account the rotor–stator interactions. It is inspired by the 3D finite element model developed in [2] for axisymmetrical rotating machines. More generally, it suggests a 3D finite element method based on modal analysis to solve linear periodically time-varying systems.

In 2 Academic rotating oscillator with asymmetric stiffnesses, 3 Analysis of Hill's equation introducing parametric quasimodes, 4 Linear stability of parametric quasimodes, the dynamic behaviour of asymmetric rotating machines is investigated through oscillators with two degrees of freedom written in complex coordinates. Indeed, oscillators with anisotropic stiffnesses in both rotating and inertial frame are useful for an introductory study of rotors with open cracks or shape imperfections. We explain the numerical method used to compute the solution of the non-autonomous governing equations. By considering Floquet solutions and Hill's infinite determinant, it is found that the dynamics of these rotating oscillators can be intrinsically defined by eigenmodes in the same way as in the autonomous case. As a consequence, free whirling and forced response are a linear combination of these parametric quasimodes. According to the linearity assumption, stability of the Floquet solution is determined by the stability of the parametric quasimodes, i.e. their decay rates.

Section 5 concerns the extension of the previous concepts to the multiple degrees of freedom governing equation arising from the 3D finite element discretization of a rotating machine with a possible shape imperfection. Due to the Hill expansion of the degrees of freedom in the frequency domain, the parametric quasimodes are computed with the modal synthesis method. The boundary conditions between substructures model the constant rotational speed Ω and the rotor–stator interactions. The last part takes advantage of the numerical tools previously introduced for the dynamic investigation of an academic test rig composed of a rectangular vertical rotor running on anisotropic supports. The comparison between numerical and experimental results gives us a first insight of the efficiency of the proposed method.

Section snippets

Influence of imperfections on the dynamic behaviour of rotating systems

The flexural behaviour of an horizontal shaft supported by anisotropic supports is represented in Fig. 1(a) [3]. The gyroscopic coupling due to possible rotating discs is neglected to focus only on the influence of imperfections. Thanks to a beam element model representing the position of the rotating shaft in an inertial frame R, a classic modal reduction can be used for stability analysis and prediction of the global dynamics [4]. The representation in the modal basis is given in Fig. 1(b)

Floquet theory and Hill's method

Thanks to the Floquet theory [8], the solution of the linear differential equation (2) can be written as the sum of n=2 independent solutionsz(t)=ZD(t)eiαt+ZR(t)eiα¯twhere both ZD(t) and ZR(t) are T-periodic functions of time with the same period as the coefficient kr(t) and α is the fundamental complex frequency and α¯ its conjugate. Unknown periodic functions ZD and ZR can be expressed by the general Fourier seriesZ(t)=j=j=+Zjeij(2π/T)twhere Zj is the harmonic contribution for the j th

Frequency lock-in mechanism

The Floquet solution being a linear combination of the parametric quasimodes, its stability is determined by their growth rates σl simply related to the eigenvalues following σl=Im(αl). Thus, the computation of eigenvalues of the undamped equations (9) truncated to a jmax order for each Ω predict the linear stability of the rotating system (Fig. 5). Referring to expressions (11), (12), the real part of αl gives informations on the frequencies of z(t) whereas a negative imaginary part triggers

Governing equations

We consider the problem of small vibrations about the equilibrium point relative to centrifugal forces of an assembly composed of a fixed part (stator) and a rotating part (rotor). As indicated by Fig. 8, the relative speed of rotation Ω is still supposed constant in magnitude and direction.

The absolute displacement u(x), velocity u˙(x) and acceleration u¨(x) of a rotating material point x (with cartesian coordinates x1, x2, x3) can be expressed in the Galilean frame R as function of their

Experimental setup

In this section, an academic test bench is studied to demonstrate the capacity of the proposed method to predict the dynamical behaviour of an asymmetric rotating machine. The experimental mockup consists in a vertical rectangular cross-section shaft mounted on an anisotropic stator. Fig. 10(a) shows a 3D modelling of the test bench. The steel structure is designed, so that flexible rotor and stator interact enough to observe important secondary harmonics in the vibratory response.

The fixed

Concluding remarks

Understanding and simulating the global dynamic behaviour of rotating machines are necessary for design and on-line monitoring. Many works have already showed that rotor axisymmetry breaking such as shape imperfections or cracks lead to linear periodic time-varying systems governed by Floquet theory. Although lot of developments have been done to simulate the whirling of asymmetric rotor, the suggested finite element models were mostly limited to beam kinematics and, if not, were not taking

Acknowledgements

The authors wish to acknowledge P. Piteau and T. Valin from the CEA Saclay for their contributions to the experimental investigations.

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