Model and Stability Analysis of a Flexible Bladed Rotor

This paper presents a fully bladed flexible rotor and outlines the associated stability analysis. From an energetic approach based on the complete energies and potentials for Euler-Bernoulli beams, a system of equations is derived, in the rotational frame, for the rotor. This later one is made of a hollow shaft modelled by an Euler-Bernoulli beam supported by a set of bearings. It is connected to a rigid disk having a rotational inertia. A full set of flexible blades is also modelled by Euler-Bernoulli beams clamped in the disk. The flexural vibrations of the blades as well as those of the shaft are considered. The evolution of the eigenvalues of this rotor, in the corotational frame, is studied. A stability detection method, bringing coalescence and loci separation phenomena to the fore, in case of an asymmetric rotor, is undertaken in order to determine a parametric domain where turbomachinery cannot encounter damage. Finally, extensive parametric studies including the length and the stagger angle of the blades as well as their flexibility are presented in order to obtain robust criteria for stable and unstable areas prediction

Stability Analysis of Beams Rotating on an Elastic Ring Application to Turbo machinery Rotor-Stator Contacts

Summary This paper presents a model of flexible beams rotating on the inner surface of an elastic stationary ring. The beams possesses two degrees of freedom, traction/compression and flexure. The in-plane deformations of the ring are considered and a single mode approximation is used. The model has been developed within the rotating frame by use of an energetic method. To better understand the phenomena occurring, the degrees of freedom of the beams can first be treated separately then together. Stability analysis show that even without rubbing, the radial degree of freedom of a beam rotating on an elastic ring can create divergence instabilities as well as mode couplings of the circular structure. When rubbing is considered, the system is unstable as soon as the rotational speed is non null. Moreover rubbing can couple the beams and the ring giving rise to mode coupling instabilities and locus veering phenomena. Finally, a comparison to a more complicated model of a flexible bladed-rotor in contact with an elastic casing shows a very good accordance with the phenomena occurring

The influence of crack-imbalance orientation and orbital evolution for an extended cracked Jeffcott rotor

Vibration peaks occurring at rational fractions of the fundamental rotating critical speed, here named Local Resonances, facilitate cracked shaft detection during machine shut-down. A modified Jeffcott-rotor on journal bearings accounting for gravity effects and oscillating around nontrivial equilibrium points is employed. Modal parameter selection allows this linear model to represent first mode characteristics of real machines. Orbit evolution and vibration patterns are analyzed, yielding useful results. Crack detection results indicate that, instead of 1x and 2x components, analysis of the remaining local resonances should have priority; this is due to crack-residual imbalance interaction and to 2x multiple induced origins. Therefore, local resonances and orbital evolution around 1/2, 1/3 and 1/4 of the critical speed are emphasized for various crack-imbalance orientations

Stability analysis and \mu-synthesis control of brake systems

The concept of friction-induced brake vibrations, commonly known as judder, is investigated. Judder vibration is based on the class of geometrically induced or kinematic constraint instability. After presenting the modal coupling mechanism and the associated dynamic model, a stability analysis as well as a sensitivity analysis have been conducted in order to identify physical parameters for a brake design avoiding friction-induced judder instability. Next, in order to reduce the size of the instability regions in relation to possible system parameter combinations, robust stability via \mu-synthesis is applied. By comparing the unstable regions between the initial and controlled brake system, some general indications emerge and it appears that robust stability via \mu-synthesis has some effect on the instability of the brake system

Stability analysis of rotating beams rubbing on an elastic circular structure

This paper presents the stability analysis of a system composed of rotating beams on a flexible, circular fixed ring, using the Routh-Hurwitz criterion. The model displayed has been fully developed within the rotating frame by use of an energy approach. The beams considered possess two degrees of freedom (dofs), a flexural motion as well as a traction/compression motion. In-plane deformations of the ring will be considered. Divergences and mode couplings have thus been underscored within the rotating frame and in order to simplify understanding of all these phenomena, the dofs of the beams will first be treated separately and then together. The dynamics of radial rotating loads on an elastic ring can create divergence instabilities as well as post-critical mode couplings. Moreover, the flexural motion of beam rubbing on the ring can also lead to mode couplings and to the locus-veering phenomenon. The presence of rubbing seems to make the system unstable as soon as the rotational speed of the beams is greater than zero. Lastly, the influence of an angle between the beams and the normal to the ring's inner surface will be studied with respect to system stability, thus highlighting a shift frequency phenomenon

Experimental and Numerical Investigations of a Dual-Shaft Test Rig with Intershaft Bearing

This paper deals with an experimental study of a dual rotor test rig. This machine, which was developed and built at the Laboratoire de Tribologie et Dynamique des Systèmes, Ecole Centrale de Lyon, will be first presented. It is composed of two coaxial shafts that are connected by an intershaft bearing and rotate independently, each one driven by its own motor. Their lateral vibrations and whirling motion are coupled by the intershaft bearing. The experimental tests consisting in run-ups and the associated measured unbalance response of the dual rotor will be investigated. The influence of the rotation of each rotor on the critical speeds and the associated amplitudes will be discussed. Moreover, this paper presents a numerical model of the dual rotor. Correlations between the experimental and numerical tests will be investigated. The objective is to be able to predict phenomena observed in experiments, starting from a rather fine numerical model

The influence of cracks in rotating shafts

In this paper, the influence of transverse cracks in a rotating shaft is analysed. The paper addresses the two distinct issues of the changes in modal properties and the influence of crack breathing on dynamic response during operation. Moreover, the evolution of the orbit of a cracked rotor near half of the first resonance frequency is investigated. The results provide a possible basis for an on-line monitoring system. In order to conduct this study, the dynamic response of a rotor with a breathing crack is evaluated by using the alternate frequency/time domain approach. It is shown that this method evaluates the nonlinear behaviour of the rotor system rapidly and efficiently by modelling the breathing crack with a truncated Fourier series. The dynamic response obtained by applying this method is compared with that evaluated through numerical integration. The resulting orbit during transient operation is presented and some distinguishing features of a cracked rotor are examined

Steady-state response of a random dynamical system described with Padé approximants and random eigenmodes

Designing a random dynamical system requires the prediction of the statistics of the response, knowing the random model of the uncertain parameters. Direct Monte Carlo simulation (MCS) is the reference method for propagating uncertainties but its main drawback is the high numerical cost. A surrogate model based on a polynomial chaos expansion (PCE) can be built as an alternative to MCS. However, some previous studies have shown poor convergence properties around the deterministic eigenfrequencies. In this study, an extended Pade approximant approach is proposed not only to accelerate the convergence of the PCE but also to have a better representation of the exact frequency response, which is a rational function of the uncertain parameters. A second approach is based on the random mode expansion of the response, which is widely used for deterministic dynamical systems. A PCE approach is used to calculate the random modes. Both approaches are tested on an example to check their efficiency