Mzaki Dakel

Sébastien Baguet
email: sebastien.baguet@insa-lyon.fr

Régis Dufour

Vibrations, Shocks and Noise

Keywords: Rotordynamics, asymmetric rotor, on-board rotor, finite element method, elastic bearing, support motion, parametric excitation

In generator and pump rotors installed in power plants, the rotating mass unbalance and the different motions of the rotor support are among the main sources of flexural vibrations. This work aims to observe the dynamic behavior of an on-board rotor subject to rigid support movements. The modeling takes into account six types of support deterministic motions (rotational and translational motions) when the kinetic and strain energies in addition to the virtual work of the rotating flexible rotor components are calculated. The finite element method is applied using the Timoshenko beam theory. The proposed on-board rotor model considers the rotary inertia, the gyroscopic inertia, the shear deformation of shaft as well as the geometric asymmetry of shaft and/or rigid disk of the rotor. By computing the Rayleigh damping coefficients, the effect of rotor internal damping is included in the study. The Lagrange's equations are used to obtain the differential equations of the rotor in bending relative to the rigid support which forms a noninertial reference frame. The equations of motion exhibit periodic parametric coefficients due to the asymmetry of the rotor and time-varying parametric coefficients due to the support rotations. In the presented applications, the rotor mounted on rigid/elastic linear bearings is excited by a rotating mass unbalance combined with sinusoidal oscillations of the rigid support. The dynamic behavior of the rotor is analyzed by means of rotor orbits and fast Fourier transforms (FFTs).

Introduction

Many industrial applications include rotating machines in which the rotor plays a dominant role. There are many studies concerning the prediction of dynamics of rotor systems mounted on elastic bearings in the case of fixed support [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF][START_REF] Genta | Dynamics of Rotating Systems[END_REF]. Some studies observed the instability of systems subjected to parametric excitations [START_REF] Dufour | Parametric instability of a beam due to axial excitations and to boundary conditions[END_REF]. Kang et al. [START_REF] Kang | Investigation on the steady-state responses of asymmetric rotors[END_REF] employed the Timoshenko beam finite elements for modeling asymmetric rotor-bearing systems. Some other works concentrated on the behavior of a rotor under the seismic excitations [START_REF] Srinivasan | Seismic analysis of a rotor-bearing system[END_REF][START_REF] Hori | Earthquake-induced instability of a rotor supported by oil film bearings[END_REF][START_REF] Suarez | Seismic response of rotating machines[END_REF]. Subbiah et al. [START_REF] Subbiah | Response of rotors subjected to random support excitations[END_REF] studied the response of rotor systems under random support excitations using modal analysis methods. Lee et al. [START_REF] Lee | A finite element transient response analysis method of a rotor-bearing system to base shock excitations using the state-space Newmark scheme and comparisons with experiments[END_REF] focused on the experimental behavior of a rotor under support shock excitation. Da Silva Tuckmantel et al. [START_REF] Da Silva Tuckmantel | An analysis on the supporting structure representative model in rotating systems[END_REF] represented the supporting structure (foundation) of a rotating system by coupled as well as uncoupled modes for calculating the system response. In [START_REF] Feng | Rotor-model-based identification of foundations in rotating machinery using modal parameters[END_REF], experimental tests have been shown for a flexibly supported undamped rigid block foundation in rotating machinery. Duchemin et al. [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF] observed the stability of a simple rotor model under a support sinusoidal rotation. Driot et al. [START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF] described the orbits of a rotor induced by a support harmonic rotational movement. El-Saeidy and Sticher [START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF] obtained the responses of a rigid rotor-bearing system subjected to rotating mass unbalance plus support harmonic excitations. Das et al. [START_REF] Das | Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion[END_REF] investigated the active vibration control of a flexible rotor system excited by mass unbalance and periodic rotational motion of the support. Among all the literature mentioned above, there are references studying support-excited rotor systems and whose few works dealt with the harmonically excited on-board rotors [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF][START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF][START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF][START_REF] Das | Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion[END_REF]. Moreover, these references concentrated on the investigation of dynamic behavior of either simple rotors or rotors supported by elastic bearings with constant damping and stiffness coefficients or rotors excited by support simple motions. As a consequence, the applications proposed in these works are not suitable for realistic ones. In this paper, an improved model is presented. Namely, an asymmetric rotor is discretized using the finite element method based on the Timoshenko beam theory, mounted on hydrodynamic bearings linearized with damping and stiffness coefficients calculated using the Reynolds equation [START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF], and excited by different motions of its support. The rotary and gyroscopic inertias, shaft shear deformation and rotor geometric asymmetry are taken into account. The linear equations of motion point out periodic parametric terms due to the rotor geometric asymmetry and time-varying parametric terms due to the support rotational excitations. In the presented examples, the rotor mounted on rigid or elastic linear bearings is subjected to rotating mass unbalance combined with support sinusoidal rotational motions. Numerical solutions are computed and analyzed by means of orbits of the rotor as well as fast Fourier transforms (FFTs).

Preliminary calculations

Three principal frames of reference shown in Fig. 1a are introduced to take into consideration the movement of the rotor rigid support. They are linked with the ground R g , the rigid support R and the moving rotor R l .

The rotational motions of the rotor support are defined by the angular velocity vector components ω x , ω y and ω z of the rigid support R with respect to the ground R g projected in the frame R. The translational motions of the rotor support are defined by the coordinates x O , y O and z O of the position vector O g O expressed in the frame attached to the support R. The Euler angles ψ, θ and φ (see Fig. 1b) allow defining the orientation of the rotor R l with respect to its support R. The angular velocity vector of the rotor R l with respect to the ground R g measured in the frame R l is defined by [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]:

g g l l l l l l T R R R x y z R R R R , , ω ω ω = + = ω ω ω (1)
where T is a matrix transpose. The components (

l x ω , l y ω , l z
ω ) are formulated as a function of (ψ,θ, φ ) and their time derivative as well as (ω x ,ω y ,ω z ). The rotor is supposed to rotate at a constant speed Ω. So the spinning angle φ is replaced by Ωt and its derivative φ ɺ by Ω (• denotes differentiation with respect to time t). Let us consider a generic point C 0 along the elastic line of the nondeformed shaft. Its coordinates in the frame R are (0,y,0). It is interesting to study the translational displacements u(y,t) and w(y,t) of the point C 0 due to bending expressed respectively with respect to the Ox and Oz axes of the frame R.

Energy and virtual work computations

The energies are measured from the ground and their terms are written relative to the frame fixed to the support.

Disk

Since the disk is considered to be rigid, only its kinetic energy T d is calculated as follows [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]:

( ) 2 1 with diag 2 2 g g g l l l d d d d d d d T R R R mo di y mo di d d m m m m m m m O R R m T I I I I I   = + = + -   v ω I ω I (2)
where m d is the mass of the disk,

( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 2 2 l l l l l l l l d d d mo x z y y di x z d d m m m O O O m T u v w I I I = + + + + + + - ω ω ω ω ω ɺ ɺ ɺ (3)

Shaft

Since the shaft is assumed to be flexible, it is characterized by the kinetic and strain energies and modeled by beam elements. The kinetic energy of a shaft can be obtained by taking a shaft elementary volume which can be considered as a disk of thickness dy. Thus the kinetic energy of a shaft has the following form [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]:

( ) ( ) ( ) 2
  = + + + + + + -       ∫ ∫ ∫ ∫ ρ ρ ω ω ρ ω ρ ω ω ɺ ɺ ɺ (4)
where ρ sh , S sh and l sh are respectively the density, the cross-section and the length of the shaft. respectively the mean inertia of the cross-section and the inertia characterizing the asymmetry of the shaft. The rigid support motion relative to the ground has no influence on the strain energy of the shaft because this energy depends only on the stresses and therefore on the shaft transverse deflection relative to the rotor support R. In addition to the bending deformation, the shear effects highlighted by Timoshenko and the geometric stiffening effects corresponding to the centrifugal stressing due to the support rotations are taken into account. The strain energy of a shaft is given by:

2 2 2 2 2 2 0 0 0 2 2 1 2 2 2 sh sh sh sh sh l l l mo mo sh S di sh sh sh sh sh S di sh sh sh E I G k S u w U dy dy E I dy y y y y y y u w G k S y y                    ∂ ∂ ∂ ∂ ∂ ∂        = + + + + - - -                    ∂ ∂ ∂ ∂ ∂ ∂                        ∂ ∂ - + - -     ∂ ∂     ∫ ∫ ∫ ψ θ ψ θ ψ θ ψ θ ( ) ( ) ( ) ( ) 0 0 2 2 2 2
    ∂ ∂    -     ∂ ∂               ∂ ∂ ∂ ∂   - + - + - + +            ∂ ∂ ∂ ∂           ∫ ∫ ∫ ∫ ψ θ Ω ρ ψ θ Ω ω ω (5)
where E sh and G sh are respectively the Young's modulus and the shear modulus of the shaft. mo sh k and di sh k are respectively the mean shear coefficient and the shear coefficient relative to the section asymmetry of the shaft.

Mass unbalance

Let us consider a concentrated mass unbalance m mu positioned at a point P mu of the disk (y mu =y d ) with a distance r mu from the shaft geometric center. Its initial angle with the Oz axis of the frame R at rest is η mu . The components of the mass unbalance translational velocity vector

Bearing

Fig. 2a shows a simple diagram of a hydrodynamic bearing which is composed of a fixed journal containing a rotating shaft. The points O and O l =C sep represent respectively the bearing center and the shaft geometric center. The radius, length and clearance of the bearing are respectively r be , l be and c be =r be -r sh where r sh is the shaft radius. At a constant rotor speed Ω and for a constant static load W r created by the rotor weight, the shaft center C sep in the bearing holds a static equilibrium position defined by the vector δ be,sep =<u be,sep ,w be,sep > R T expressed in the frame R or equivalently by the eccentricity be be ,sep e = δ of the shaft center in the journal and the attitude angle φ be between the W r load direction and the line of centers OC sep . In the present study, the short bearing theory is considered (l be /d be ≤1/8 where d be =2r be ) and the static solution can be obtained using the following formulations deduced from the Reynolds equation with the Gümbel boundary conditions [START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF]:

( ) ( ) ( ) ( ) 1 2 2 2 2 2 2 2 2 2 16
1 1 with 4 1 be be be be be be r be be be be be be be

r l W r l tg c d ε ε π ε ε π µ Ω ϕ ε ε + - -     = =     -     (7)
where ε be is the relative eccentricity (ε be =e be /c be ) and µ is the fluid film dynamic viscosity. The nonlinear Eq. ( 7) is solved by an iterative Newton-Raphson method and provides the eccentricity ε be and then the static radial displacement e be of the shaft center. The components of the vector δ be,sep are obtained by a classical change of basis.

The hydrodynamic fluid forces F be =<F be u ,F be w > R T produced by the bearings and expressed in the frame R can be obtained by integration of the fluid film pressure (Reynolds equation in the dynamic regime) over the bearing. In order to apply the Lagrange's equations, the virtual work δW be of these forces has to be established:

( ) T be be be be be W , = F δ δ δ δ δ ɺ (8)
If the lateral dynamic displacements δ be =<u be ,w be > R T of the shaft elastic line are assumed to be small in the vicinity of the static position δ be,sep , the linear analysis can be used by constructing a first order Taylor expansion of the fluid film forces where c be and k be are the damping and stiffness matrices of the linearized hydrodynamic bearing (see Fig. 2b) whose analytical expressions can be found in [START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF].

c c k k c c k k     = =         c k (10) (a) (b)

Equations of motion

The finite element method is chosen for discretizing the rotor and describing its flexural motion as a function of the nodal displacement vector defined by δ n =<u n ,w n ,θ n ,ψ n > R T , i.e., the rotor has two translations and two rotations at each node. The finite element used for the shaft modeling has two nodes and the shape functions are based on the Timoshenko beam theory. The linear equations of motion of the finite element rotor model are obtained after applying the Lagrange's equations to the energies for the disk, the shaft finite element and the mass unbalance as well as to the virtual work of the hydrodynamic bearing and assembling appropriately the produced matrices and vectors. They are written with respect to the noninertial reference frame connected to the rotor rigid support R:

( ) ( ) ( ) ( ) ( ) r r r r r r r be r ,sep be r ,sep t t t t , + + = + + M δ C δ K δ F F δ 0 K δ ɺɺ ɺ (12)
where M r (t), C r (t) and K r (t) are the parametric matrices of mass, damping and stiffness of an asymmetric rotating rotor on moving support. r δ ɺɺ , r δ ɺ and r δ are the acceleration, velocity and displacement vectors of dimension 4(n esh +1)×1 hence n esh is the number of shaft finite elements. F r (t) is the external force vector. F be (δ r,sep ,0) is the hydrodynamic force vector caused by the bearing in the static case which opposes the rotor weight r W d ,sh F . In addition, δ r,sep is the static solution vector of the rotor due to the hydrodynamic bearing.

The matrices of Eq. ( 12) are expressed in what follows:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
Ω Ω Ω Ω Ω Ω Ω ω Ω Ω ω Ωω = + + = + + + + + = + + + + + + M M M M C C C C C C C K K K K K K K K ɺ ɺ ( ) ( ) 2
+ + + + + K K K K K K K K K K K ɺ ɺ ( ) ( ) ( ) ( )
+ + + + + K K K K K ɺ ɺ (13)
The subscripts "d", "sh", "be" and "su" refer respectively to the disk, shaft, bearing as well as support and express the contribution to the phenomenon represented by the corresponding matrix. The superscripts "c 2 " and "s 2 " denote the geometric asymmetry of the rotor expressed in terms of the time-varying trigonometric functions cos(2Ωt) and sin(2Ωt). The superscript "id" stands for the rotor internal damping introduced by estimating the Rayleigh damping coefficients, "g" for the rotor gyroscopic effect, "e" for the shaft elasticity corresponding to the bending and shear deformations, "re" for the rotational effects due to the support rotations (these effects come from the kinetic energies of the disk and the shaft) and "gse" for the geometric stiffening effects corresponding to the centrifugal stress due to the support rotations.

The vector F r (t) is defined as follows:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2
+ + + + + F F F F F F F V V V V V V V V V ɺ ɺ ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 2 2 1 1 1 1 1 1 2 2 2
+ + + + + + - + - + + - - - + - + - V V V V V V V V ɺ ɺ ɺ ɺ ɺɺ ɺ ɺ ɺ ɺ ɺɺ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
y x z cos t θ ψ θ ψ θ ψ ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω Ωω ω ω Ωω ω ω ω Ωω ω ω Ω ω Ωω ω + + - - - + - + + - - + - - - + + + - - - + + + V V V V V V V V ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 ,s x y x z y z d ,sh ,su ,s z x x y d ,sh ,su cos t sin t sin t ψ θ ω Ω ω Ωω ω ω Ω ω Ωω ω ω Ω + - - + + + V V ɺ ɺ (14)
where V mu (F mu (t)), V d,sh,su (F d,sh,su (t)) and V mu,su (F mu,su (t)) are the load vectors (force vectors) associated respectively with the mass unbalance, the inertia force due to support motions and that due to coupling between both phenomena. The superscripts "c 1 " and "s 1 " signify the components of the mass unbalance force expressed in terms of the timevarying trigonometric functions cos(Ωt) and sin(Ωt). The superscripts u, w, ψ and θ denote the direction of the action force components associated with the rotor support motions.

The transient dynamic motion of the rotor is then obtained by solving Eq. ( 12) by means of the implicit Newmark scheme based on the average acceleration. The static equilibrium position δ r,sep is used to initialize the transient dynamic problem. The final integration time is chosen such that the transient effects have disappeared and the steady-state regime has been reached.

Results and discussion

The symmetric rotor-rigid/elastic linearized bearing system presented in Fig. 3 is assumed to be subjected to rotating mass unbalance as well as to support harmonic rotation around the Ox axis given by ω x =ω x,a cos(Ω x t) in rad/s. The physical properties as well as the geometry of the rotor and the bearings are given in Table 1. The shaft is discretized into eight equal length 2-node Timoshenko beam finite elements. The disk is located at node 5 and the bearings # 1 and # 2 are respectively placed at nodes 1 and 9. The rotor is operated at a constant speed of rotation Ω=1200 rpm (20 Hz=mass unbalance frequency). The static equilibrium position of the shaft geometric center in the fluid film bearings is given by δ be,sep =<-5.71×10 -5 ,-1.76×10 -4 > R T m. The bearing damping and stiffness matrices are expressed in what follows:

    × × × × = =     × × × ×     c k (15)
The rotor system is supported either by rigid bearings or hydrodynamic bearings leading in the case of fixed support either to symmetric damping and stiffness matrices with isotropic diagonal and cross-coupling terms or to nonsymmetric matrices respectively. Fig. 4 shows the classical disk orbit due to the mass unbalance for rigid bearings and a fixed support. Since the matrices of the rotor-bearing system are symmetric and skew-symmetric with isotropic diagonal and cross-coupling components, the dynamic behavior is symmetric and the orbit is circular. Its center coincides with the point O (bearing center). FFT shows that the rotor displacement has an amplitude almost of order 1×10 -7 m and the same mass unbalance frequency (20 Hz). ɺ in Eq. ( 14)). The two transverse displacements associated with these force components are different and this breaks the asymmetry of the rotor behavior. For the selected speed of rotation of the rotor, the z-amplitude is very higher than the x-amplitude which remains mostly the same. FFT exhibits two frequency components due to the mass unbalance excitation (20 Hz) and to the support harmonic motion (80 Hz). Fig. 6 presents the disk orbit due to the mass unbalance effect in the presence of hydrodynamic bearings. The nonsymmetric damping and stiffness coefficients make the bearings anisotropic and the orbit elliptical with diagonal axes defining the phase between the mass unbalance excitation and the rotor response. The orbit center coincides with the point C sep (static position of the shaft center in the bearings). The orbit is large compared to that corresponding to rigid bearings because of the combination between the bending modes of the rotor and the rigid body modes relative to the rotor motion in the bearings. z-FFT indicates that the disk vibration has an amplitude of about 7×10 -6 m and the same mass unbalance frequency (20 Hz). 7 displays the disk orbits for hydrodynamic bearings during support rotational motions. The orbit characteristics (shape and magnitude) change with amplitude and frequency of the support harmonic excitation. For very small amplitudes and different frequencies, the orbit shapes become more complicated with respect to those obtained when the support is fixed (see Fig. 7a). It should be noted that the dynamic behavior is all the more modified as the frequency of the support excitation is high compared with the natural frequencies or the speed of rotation of the rotor. On the other hand, the support motion amplitudes change the x and z orbit magnitudes. FFTs exhibit two frequency components due to the mass unbalance excitation (20 Hz) and to the support harmonic motions (50 Hz and 80 Hz). Even if the support rotations have an influence on the mass unbalance excitation (see Eq. ( 14)), this influence does not appear clearly in FFTs of the rotor motion during the support harmonic rotation because its amplitude is very small compared to the speed of rotation of the rotor (mass unbalance frequency).

C sep

Conclusions

A finite element model is presented to analyze the dynamic behavior of a symmetric on-board rotor whose support is subjected to sinusoidal rotation. The rotational effects and the geometric stiffening effects relative to the centrifugal stressing due to the support rotations are taken into account. The support rotations create time-varying parametric coefficients which can lead to lateral dynamic instability. In the case of a rotor mounted on rigid bearings and running at the lower speeds of rotation, the support rotation around a transverse axis creates an orbit having its greatest magnitude in the perpendicular transverse direction. Unlike the previous case, the support rotation effects concern the two transverse directions when the rotor is carried by hydrodynamic bearings. It is noted that the shape and the magnitude of the orbits can be significantly affected by the support motion frequency and amplitude respectively. In the case of considerable rotation amplitude compared to the speed of rotation of the rotor, it is shown that the mass unbalance forces can depend on the support rotation around a transverse axis. The frequency components due to the mass unbalance excitation and to the support harmonic motions appear in FFTs of the rotor flexural vibrations. In the case of large orbits, the assumption of a linearized hydrodynamic bearing model is questionable. A nonlinear model is to be considered and the hydrodynamic bearings have to be treated as external nonlinear forces acting on the shaft.

I

  are used to distinguish the effects due to the mean inertia of the disk mass and those due to the inertia modeling the disk geometric asymmetry. In addition, d y m I is the inertia of the disk mass about the Oy axis.

Fig. 1 .

 1 Fig. 1. (a) Frames of reference for the on-board rotor, (b) Euler angles.

v

  are functions of (ω x ,ω y ,ω z ) as well as (x O ,y O ,z O ) and are used in the kinetic energy which characterizes the mass unbalance[START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]:

Fig. 2 .

 2 Fig. 2. (a) Sketch of a hydrodynamic bearing, (b) Bearing damping and stiffness coefficients.

Fig. 3 .

 3 Fig. 3. Description of the on-board rotor system for numerical examples.

Table 1 .

 1 Details of the investigated rotor system with hydrodynamic bearings Density of both disk and shaft material Radius, thickness and location of the disk shaft material (E sh , ν sh ) Radius and length of the shaft Mass unbalance 7800 kg/m 3 0.15 m, 0.03 m, 0.2

Fig. 4 .

 4 Fig. 4. Mass unbalance orbit and z-FFT of the rotor lateral displacements at the disk for rigid bearings and fixed support.

Fig. 5

 5 Fig. 5 gives the disk orbit and FFT of rotor vibrations due to the mass unbalance and the support rotation around the Ox axis. This rotation generates vectors corresponding to force components acting respectively in the Ox

O

  direction (due to the Coriolis acceleration, i.e., the term Eq. (14)) and in the Oz direction (due to the tangential acceleration, i.e., the terms

Fig. 5 .

 5 Fig. 5. Disk orbit and z-FFT of the flexural vibrations for rigid bearings and a support harmonic rotation: ω x,a =5×10 -2 rad/s and Ω x =80 Hz.

Fig. 6 .

 6 Fig. 6. Mass unbalance orbit and z-FFT of the rotor transverse displacements at the disk for flexible bearings and fixed support.

Fig.

  Fig.7displays the disk orbits for hydrodynamic bearings during support rotational motions. The orbit characteristics (shape and magnitude) change with amplitude and frequency of the support harmonic excitation. For very small amplitudes and different frequencies, the orbit shapes become more complicated with respect to those obtained when the support is fixed (see Fig.7a). It should be noted that the dynamic behavior is all the more modified as the frequency of the support excitation is high compared with the natural frequencies or the speed of rotation of the rotor. On the other hand, the support motion amplitudes change the x and z orbit magnitudes. FFTs exhibit two frequency components due to the mass unbalance excitation (20 Hz) and to the support harmonic motions (50 Hz and 80 Hz). Even if the support rotations have an influence on the mass unbalance excitation (see Eq. (14)), this influence does not appear clearly in FFTs of the rotor motion during the support harmonic rotation because its amplitude is very small compared to the speed of rotation of the rotor (mass unbalance frequency).

Fig. 7 .

 7 Fig. 7. Disk orbits and z-FFTs of the lateral vibrations for flexible bearings and two amplitudes of support harmonic rotations: ω x,a = (a) 10 -2 rad/s, (b) 5×10 -2 rad/s, (Ω x =20 Hz, 50 Hz AND 80 Hz).

  ω x,a =5×10 -2 rad/s (a) ω x,a =10 -2 rad/s


Loïc Peletan
email: loic.peletan@insa-lyon.fr.

Mohamed Torkhani

Georges Jacquet

Ludovic May

Sébastien Baguet

Philipe Voinis

Georges Jacquet-Richardet

Vibrations, Shocks and Noise Experimental and transient/harmonic numerical investigations for the risk assessment of a rotor response after blade

Keywords: rotordynamics, blade-off, rub-impact, experimental test bench, transient simulations, harmonic balance method

In nuclear power plant turbosets, the design-basis accident consists of a blade the accidental shutdown, a severe rotor-casing contact interaction may occur at critical speeds due to large shaft line displacements originated by a high imbalance excitation. EDF must be able to check that the bearings are capable of withstanding the loads developed during passing through critical speeds in the deceleration phase without catastrophic consequences for th shaft line. The targeted non-linearity is due to frictional rotor EURoPE test rig. This test bench has been designed for experimental investigation of rotor shutdown. The second focus of the paper is on a modeling framework tailored to simulating the overall dynamic behavior of a non linear rotor subjected to imbalance. A 1Dequations due to contact conditions are solved through an explicit multiplier approach dealing with a node-to-line contact strategy used for fast assessment of the model steady-state

Introduction

Industrial context and problem statement

Turbomachines for aeronautical and terrestrial appli of which forms a rotor and a non-rotating bladed disks assembly, named the stator. In turbo requirement of higher efficiency has reduced the clearances between shaft (rotating part) The safety of turbomachines involves the control of risk situations such as contact between rotating and stationary parts of the turbine, usually referred to as rubbing (Muszynska [START_REF] Muszynska | Rotor to stationary element rub related vibration phenomena in rotating machinery -literature survey[END_REF], Beatty [START_REF] Beatty | Differentiating rotor response due to radial rubbing[END_REF]), due to "low unbalance" and accidental situations referred to as "high unbalance". In the first situation, intermittent contacts of low levels will be the location for energy exchange between the rotating part and the fixed part. Various modes can then occur, basis accident consists of a blade-off on the low pressure turbine last stage. During casing contact interaction may occur at critical speeds due to large shaft line riginated by a high imbalance excitation. EDF must be able to check that the bearings are capable of withstanding the loads developed during passing through critical speeds in the deceleration phase without catastrophic consequences for th linearity is due to frictional rotor-stator contact. The first focus of the paper is EDF's modular EURoPE test rig. This test bench has been designed for experimental investigation of rotor-stator contact during emergency focus of the paper is on a modeling framework tailored to simulating the overall dynamic behavior of a -based FE model is used for the entire turbine system. The ions are solved through an explicit time-marching procedure combined with the Lagrange line contact strategy. On the other hand, the Harmonic Balance Method (HBM) state response.

impact ; experimental test bench ; transient simulations ; harmonic balance method

Industrial context and problem statement

Turbomachines for aeronautical and terrestrial applications mainly comprise rotating bladed disks, the assembly rotating bladed disks assembly, named the stator. In turbo requirement of higher efficiency has reduced the clearances between shaft (rotating part) and stator (stationary part). The safety of turbomachines involves the control of risk situations such as contact between rotating and stationary parts of the turbine, usually referred to as rubbing (Muszynska [START_REF] Muszynska | Rotor to stationary element rub related vibration phenomena in rotating machinery -literature survey[END_REF], Beatty [START_REF] Beatty | Differentiating rotor response due to radial rubbing[END_REF]), due to "low unbalance" and accidental situations referred to as "high unbalance". In the first situation, intermittent contacts of low levels will be the location for energy exchange between the rotating part and the fixed part. Various modes can then occur, Experimental and transient/harmonic numerical investigations for off with radial

Philipe Voinis b , off on the low pressure turbine last stage. During casing contact interaction may occur at critical speeds due to large shaft line riginated by a high imbalance excitation. EDF must be able to check that the bearings are capable of withstanding the loads developed during passing through critical speeds in the deceleration phase without catastrophic consequences for the stator contact. The first focus of the paper is EDF's modular stator contact during emergency focus of the paper is on a modeling framework tailored to simulating the overall dynamic behavior of a

The highly nonlinear marching procedure combined with the Lagrange Harmonic Balance Method (HBM) is also impact ; experimental test bench ; transient simulations ; harmonic balance method cations mainly comprise rotating bladed disks, the assembly rotating bladed disks assembly, named the stator. In turbo-machinery, the and stator (stationary part). The safety of turbomachines involves the control of risk situations such as contact between rotating and stationary parts of the turbine, usually referred to as rubbing (Muszynska [START_REF] Muszynska | Rotor to stationary element rub related vibration phenomena in rotating machinery -literature survey[END_REF], Beatty [START_REF] Beatty | Differentiating rotor response due to radial rubbing[END_REF]), due to "low unbalance" and accidental situations referred to as "high unbalance". In the first situation, intermittent contacts of low levels will be the location for energy exchange between the rotating part and the fixed part. Various modes can then occur, depending on the speed of rotation of the rotor, the loading and the geometrical and physical parameters of the system. These modes can lead to the loss of contact between the rotor and the stator, but also to unstable phenomena, the consequences of which can prove to be disastrous. The second configuration or high unbalance will lead to severe contact between the rotor and the stator. Rub is usually caused by unbalanced rotor, blade-off, misalignment of the rotor centerline, thermal unbalance, casing excitations. This can occur in the start up procedures of a new machine or of an older one after a shutdown and must be taken into consideration in the design of the machine. In this study, the shaft-diaphragm interaction only is considered: indeed, in real turbosets, the blades-todiaphragm contact is negligible in comparison to the shaft-stator interaction.

To do this, rub experiments were performed on the "EURoPE" modular test rig of EDF R&D. The test rig has been designed in order to be as much as possible representative of industrial machines, mainly steam turbines of power plants. This rig has been employed for several researches in the field of rotor dynamics (see Stoisser et al. [START_REF] Stoisser | A comprehensive theoretical, numerical and experimental approach for crack detection in power plant rotating machinery[END_REF]). In the first rig configuration of the shaftline without inertia flywheels (see [START_REF] Torkhani | Light medium and heavy partial rubs during speed transients of rotating machines: Numerical simulation and experimental observation[END_REF]), the shutdown transient and rubbing durations were very short (respectively less than 50 and 10 seconds). These durations did not enable us to investigate all the physical phenomena. Thus, it was decided to go on with the rubbing experiments in order to ensure through a more representative configuration (EURoPE shaftline with inertia flywheel) whether the Rotor-Stator contact generates or not complex and uncontrolled phenomena. The current configuration allows for longer contact and shutdown durations. The phenomenological analyses will be undertaken on the basis of the test results in this configuration. Indeed, the presence of the inertial flywheel can a priori imply the following effects:

• amplification of the gyroscopic effects; • variation of the shaft deformation across the rotating speed at which the contact occurs;

• decrease of the critical speed;

• possibilities of persistent Rotor-Stator contact (leading to full annular rubbing) and backward whirling motion.

Article focus

In order to verify the foregoing physical explanations of rubbing phenomena, theoretical predictions and experimental observations were performed. A test rig has been designed to allow for rubs to a rotating shaft during start-up and shutdown operating conditions. A description of the experimental rig and the procedures used to investigate the dynamical responses of the system are presented. This small test rig cannot be fully representative of the behavior of a big sized real machine. However scaling effects have been considered. It is then expected that some valuable vibration features of the rub-impact rotor system can be observed and the research can provide some useful information for the fault diagnostics of rotating machinery.

The experiments were performed under different conditions to observe the nonlinear vibrations. For instance, a gradual unbalancing of the rotating shaft is used to generate rubs of different intensities. Some quantities have been measured: shaft vibrations in the section where contact occurs, forces transmitted to the supporting structure, duration of contact and angular decelerations.

Investigations of the motion of the rotor with rubbing experimental methods in the case of 'large' are presented in this paper. The general framework for numerical investigations is also presented.

Description of the experimental setup

Description of the EURoPE test rig

The bench is a two-fluid bearing horizontal rotor system (see Figure 1). The flexible shaft is composed of three rotors, with a central modulus that can be modified or replaced in order to study different contact samples. The overall shaft length is 3150mm with a nominal diameter of 210mm. The shaft is connected to an electrical motor, driven by inverter, by means of a flexible coupling. The total shaft mass is approximately 450kg and the main inertia disk has a mass of about 250kg. The first and second bending mode frequencies of the rotor system are respectively close 17.3Hz and 19.4Hz at nominal rotating speed (1500rpm). The first torsion mode is equal to 84Hz. The motor drives the shaft from 0rpm up to a maximum design rotating speed of 3000rpm, but tests are generally carried out at lower speeds, because the aim is to analyze the behavior in the neighborhood of the first flexural critical speed under safe conditions. The shaft is supported by two equal lemon shaped oil-film bearings and by two bearing housings, connected to an oil circulation system to lubricate the bearings. The supporting structure can be considered as rigid in the considered speed range. The bearings are equipped with two proximity probes in the horizontal and vertical directions for the measurements of relative shaft-journal vibrations. The casings are equipped with two accelerometers for measuring its vibrations in vertical and horizontal directions. Probes, located near each bearing and near the contact position, measure the shaft vibrations in vertical and horizontal directions.

In order to better simulate the real process of the rub, we have designed a special structure of stator that can make it possible to perform a safe and reliable contact experiment, as shown in Fig. 2.

Motor

Bearing 1 Inertia 2 Inertia 3 Bearing 2 Inertia 1

Figure 2 View of the stator and the contact zone It is easy for the stator to be mounted, disassembled or interchanged. Not only can the concentricity of the stator and the rotor be adjusted, but also the clearance is adjustable by changing the different inner stators. In fact, different materials (respectively Steel, Teflon and Brass) can be used for the inner part of the stator in order to meet different experimental conditions. However, this leads to some difficulties to have a valid reference case. In fact, by considering two rundowns of the rotor with and without mounted stator, small differences in the unbalance sensitivity of the shaftline might be introduced when the stator is assembled to or disassembled from the whole structure. The stator structure is clamped to a massive concrete block by means of four bolts.

Force sensors are installed between the contact interface and the casing in order to measure the forces generated by the contact interaction.

Tests were carried out to check the effectiveness of the sensor system to measure the contact forces.

Validation of the unbalance procedure

Our primary objective is to study the rubbing phenomena only during the accidental shutdown. Then, a dedicated device was designed to guarantee no-rubbing conditions during the speed-up of the shaftline ("well-balanced" shaftline). The idea consists in the excitation of the shaft by unbalance force only during rundown. The principle relies on installing on the stator a case in which one places a known mass made up of water ice block. The mass of ice block corresponds to the unbalance which one wishes to impose on the machine during the rundown. The case (called "IceCase") is in correspondence of the second inertia disc with a mass eccentricity of 105mm and an angular position of 30° compared to the Top-Tour (phase reference). To balance the machine during the run-up, one has to install a compensation mass (called "CMass") in phase opposition with the "IceCase" (i.e. with a 210° angular position). These positions were found experimentally in order to make it possible to maximize the vibration amplitude of the shaftline in the contact zone. Actually, one must manage to have in-phase vibrations due to the unbalance and to rotor bow over the first critical speed. Since this varying unbalance must not exceed a maximum authorized value (security issue), a fixed unbalance of 8.3g is placed in correspondence of the first inertia disc with a mass eccentricity of 250mm and an angular position of 345°. This second unbalance aims at "helping" the balancing procedure. Figure 3 represents the prototype.

IceCase CMass

Figure 3 Ice block case and compensation mass installed on the second inertia disc of the test rig Thus, the "well-balanced" shaft is driven by the motor to run up until the rotating speed of 1525rpm. This guarantees that the rubbing will not occur until the ice has completely melt (larger unbalance amount generates larger vibration amplitudes). During heavy contact tests, the stabilization speed was raised to 1640rpm. Once this rated speed reached, one has to wait for complete ice melting, i.e. until full stabilization of the vibration levels. The shaftline is then subjected to an unbalance corresponding to the "CMass" mass. One cuts then the engine supply off in order to allow for shaftline free rundown. The vibration of the shaftline is then directly correlated with the compensation mass.

Tests with unbalances (applied to the shaft balancing plane) have been run in order to check the sensitivity of the bearing response to unbalancing forces and to identify all resonances of the system. This model and its results were used to plan the experiments and to aid in the balancing procedures.

Results show that the unbalance device is fully operational and that the force sensors assembly allows for measuring the instantaneous forces. The test rig is then suitable for rubbing tests to be performed.

The following characteristics have been measured on the bench: rotor-stator radial clearance equal to 400µm, radial rotor eccentricity equal to 105µm and radial rotor bow equal to 75µm. When radial deflection exceeds the remaining clearance between the external surface of the whirling shaft and the internal surface of the stationary part, then contact occurs.

Analysis of experimental results -case of heavy contact

After setting up the rig, some first experimental results have been obtained. In order to operate in safe conditions towards the contact severity, we adopted a gradual modification of inner stator material (Teflon, Brass then Steel) and a gradual increase in the unbalance amount. Moreover, since rubbing is a kind of nonlinear phenomena, the experimental results cannot be the same at every time. In this respect, several experiments for a set-up need to confirm a peculiar physical phenomenon. Through these several experiments the noise and the signal can be manifested. In our case, tests were carried out several times in order to ensure a "repeatability" of the experimental results. Doing this, we can avoid signals noise while also controlling all the procedures of the set-up.

Results obtained during the rundown will be shown in terms of shaft orbits, contact duration, contact force and rotational speed variation. Values recorded by force sensors represent only roughly the contact forces because the measured forces are affected by the inertia forces of the ring support. Orbits in presence of rub are obtained also in bearings but are omitted here for clarity.

The unbalance mass amount used to simulate heavy contact is 77.15g, which corresponds to a total applied unbalance of 0.0017Kg.m on disc 1 plus 0.0081Kg.m on disc 2. The corresponding shaft vibrations amplitudes are plotted in Fig. 4. Two main events are observed from the vibration measures. They correspond to:

• appearance of shocks in the signal measured by the force sensors. The contact initiation is recorded by the sensor beginning from 1173rpm (right cursor in Fig. 4).

• extinction of the shock from the force sensor after 964rpm (left cursor in Fig. 4). The time interval during which contacts happen is more than a quarter of the period of revolution. As previously specified the rotational speed has also been measured with a Top-Tour, which by numerical time derivation gives the angular accelerations. The angular velocity and deceleration graphs are shown in Figure 6. The measured rotation speed curve depicts clearly a visible modification with the naked eye at the shock time. The angular deceleration graph shows:

• a first phase of null acceleration corresponding to the shaft whirling at the stabilized speed, • a second phase of strictly negative acceleration corresponding to the natural deceleration of the whirling shaft in absence of rubbing, • a third phase of slope break in the deceleration curve corresponding to the shock.

Besides the natural deceleration, the friction torque acting on the rotor has the effect to decelerate it.

Modeling and simulation results

The EURoPE test rig model

A 1D finite element numerical model of the test rig is developed in order to run numerical simulations. It is displayed in Fig. 7. The shaftline rests on 2 bearings, the characteristics of which are modeled by linear stiffness and damping coefficients. A simplified model of the foundation is taken into account at the bearing positions: mass, stiffness and damping coefficients.

COUPLER TELMA BEARING BEARING

Bearing probes 1&2

The experimental identification of rotor modes is carried out at different rotating speeds in the vicinity of the first critical speed. The resonance rotating speeds are respectively equal to 1053rpm (mode 1) and 1148rpm (mode 2).

To validate the 1D numerical model of the test rig mass and stiffness model parameters are first updated considering the shaft in free-free conditions. The stiffness and damping coefficients of the bearings and the flexible coupling are determined and then adjusted in order to obtain the experimental vibration level related to the wellbalanced shaftline. Eigen frequencies and reduced damping values obtained using the updated FE model are successfully compared to the corresponding experimental values. The values of frequency show very good accordance between numerical and experimental results. Residual damping is also well reproduced by the numerical model, whilst a discrepancy of the first mode residual damping is observed for low and high rotating speeds (i.e. away from the critical speed). The accuracy of the shaftline model is thus verified at least in the vicinity of the first critical speed.

In order to predict the behavior of the bench, two different numerical methods will be used and their results will be shown during the presentation. The two methods are described hereafter. They both solve the equation of motion resulting from the space FEM discretisation of the rotor (see Roques et al. [START_REF] Roques | Modelling of rotor speed transient with rotor to stator contact[END_REF] for more details):

ሷܙۻ ሺݐሻ + ۱ܙሶ ሺݐሻ + ۹ܙሺݐሻ + ൫ܙሺݐሻ൯ -ܘሺݐሻ = (1)
where ܙ represents the displacement vector for all of the ݊ degrees of freedom (DOFs); ۹, ۱ and ۻ are the generalized ݊ × ݊ stiffness, damping and mass matrices; is the nonlinear force vector and ܘ is the external excitation force vector. As a general statement ۱ = ߮ሶ ۵ + ۯ bearings , where ۵ is the gyroscopic matrix, ۯ ୠୣୟ୰୧୬ୱ is the bearings damping matrix and ߮ is the angular position of the shaftline. ۹ = ۹ shaft + ۹ bearing + ߮ሷ ۹ stiff where ۹ shaft is the shaftline stiffness matrix, ۹ bearing is the bearings stiffness matrix and ۹ stiff is the stiffness matrix due to angular velocity variations.

ܘሺݐሻ = ݉ ݎ ሾሺ߮ሷ sin߮ + ߮ሶ ଶ cosφሻܘ ଵ + ሺ-߮ሷ cos߮ + ߮ሶ ଶ sin߮ሻܘ ଶ ሿ ே unbalance ୀଵ
where ݉ is the unbalance mass and ݎ is the unbalance excentricity. ܘ ଵ and ܘ ଶ are unit vectors.

Harmonic Balance Method for fast assessment of the system behavior

The Harmonic Balance Method (HBM) is a well known technique, used to compute periodic solutions for dynamic systems. This method consists in solving the equations of movement in the frequency domain, rather than in the time domain. As the movement is supposed periodic, the transient effects are not taken into account. Thus, the angular velocity is considered constant ሺ߮ሷ = 0ሻ in the above equations.

When the external excitations are periodic, it is valid to assume that a steady state solution for Eq. ( 1) exists, and that this solution is also periodic. The displacements can thus be written as a truncated Fourier series:

ܙሺݐሻ = ۿ + ሺۿ ଶିଵ cos൫݉ ߱ݐ൯ + ۿ ଶ sin൫݉ ߱ݐ൯ሻ ே ୀଵ (2)
where ۿ , ݅ ∈ ሾ0 … 2ܰሿ, are the Fourier coefficients of ;ܙ ߱ is the fundamental pulsation of the external excitation, and ݉ , ݆ ∈ ሾ0 … ܰሿ are positive integers.

Similarly, ሺܙሻ and ܘሺݐሻ can be rewritten using the same form:

ܘሺݐሻ = ۾ + ሺ۾ ଶିଵ cos൫݉ ߱ݐ൯ + ۾ ଶ sin൫݉ ߱ݐ൯ሻ ே ୀଵ (3)
ሺݐሻ = ۴ + ሺ۴ ଶିଵ cos൫݉ ߱ݐ൯ + ۴ ଶ sin൫݉ ߱ݐ൯ሻ ே ୀଵ (4)
As described in [START_REF] Petrov | Analytical formulation of friction interface elements for analysis of nonlinear multiharmonic vibrations of bladed disks[END_REF] Eqs. (2) (3) and ( 4) can be substituted into Eq. ( 1), following which a Galerkin procedure is applied to transform the nonlinear differential Eq. ( 1), of dimension ݊, into an algebraic nonlinear system of equations, of dimension ݊ HBM = ݊ሺ2ܰ + 1ሻ:

,ۿሺ܀ ߱ሻ = ܈ሺ߱ሻۿ + ۴ሺۿሻ -۾ =

where:

܈ = diag ൬۹, ۹ -݉ ଵ ଶ ߱ ଶ ۻ ݉ ଵ ߱۱ -݉ ଵ ߱۱ ۹ -݉ ଵ ଶ ߱ ଶ ۻ ൨ , ⋯ , ۹ -݉ ே ଶ ߱ ଶ ۻ ݉ ே ߱۱ -݉ ே ߱۱ ۹ -݉ ே ଶ ߱ ଶ ۻ ൨൰ ۿ = ሾۿ ் , ۿ ଵ ் , ⋯ , ۿ ଶே ் ሿ ் , ۴ = ሾ۴ ் , ۴ ଵ ் , ⋯ , ۴ ଶே ் ሿ ் and ۾ = ሾ۾ ் , ۾ ଵ ் , ⋯ , ۾ ଶே ் ሿ ்
are the vectors of the Fourier coefficients for displacements, nonlinear forces and external excitations, respectively. Equation ( 5) needs to be solved for .ۿ As this equation is still nonlinear, an appropriate nonlinear solver must be used to derive a correct solution. In the present situation, a Newton-Raphson solver is used. The Newton-Raphson solves Eq. ( 5) in an iterative manner :

ۿ ሺାଵሻ = ۿ ሺሻ + ቆ ۿ൫܀߲ ሺሻ ൯ ۿ߲ ቇ ିଵ ۿሺ܀ ሺሻ , ߱ሻ
It has been demonstrated that the combined use of HBM and a Newton-Raphson solver is equivalent to the Incremental Harmonic Balance Method (IHBM) [START_REF] Lau | A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems[END_REF]. An Alternating Frequency Time (AFT) algorithm [START_REF] Cameron | An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems[END_REF] was used for the computation of ۴ሺۿሻ, and as the relationship between the displacements and the nonlinear forces is known only in the time domain (i.e. is a priori unknown in the frequency domain), direct and inverse Fourier transforms must be used to determine ۴ሺۿሻ. The displacements ሺۿሻ in the frequency domain are transferred to the time domain using an inverse Fourier transform. Once the time domain displacements ሺܙሻ are known, the (time-domain) nonlinear forces ሺሺܙሻሻ can be calculated. A direct Fourier transformation is then performed, to obtain the nonlinear forces in the frequency domain ሺ۴ሻ. The next Newton-Raphson iteration can then be carried out. In practice, fast Fourier transform algorithms (FFT) are used due to their combined speed, accuracy and robustness. This procedure can be summarized as follows:

ۿሺ߱ሻ fft షభ ሱۛሮ ܙሺݐሻ → ሺݐሻ fft → ۴ሺ߱ሻ
Similarly, the Jacobian term, డ܀ሺۿ ሺೖሻ ሻ డۿ , is calculated via analytical derivation rather than through the use of finite differences, in order to ensure better accuracy and faster convergence. A pseudo-arc length continuation method [START_REF] Sundararajan | An algorithm for response and stability of large order non-linear systemsapplication to rotor systems[END_REF] is used for the parametric study.܀ሺ,ۿ ߱ሻ = ܈ሺ߱ሻۿ + ۴ሺۿሻ -۾ =

Numerical time integration strategy

The second method solves Eq. (1) using a time integration scheme. The transient effects are taken into account. The angular deceleration which is originated by contact between the shaftline and the stator (considered as an unknown) is taken into account. So an additional equation computes the instantaneous rotating speed depending on the contact forces.

The highly nonlinear equations due to contact conditions are solved through an explicit prediction-correction time-marching procedure combined with the Lagrange multiplier approach dealing with a node-to-line contact strategy. In our study, a Finite Central Differences scheme combined with Lagrange multipliers (predictioncorrection algorithm) properly satisfies the contact detection and ensures the compatibility of the speed and the acceleration. The chosen time step ݐ∆ is equal to10 -6 s, so that convergence of the quantities of interest with respect to the time step is achieved.

Conclusion

The emphasis of this paper was twofold: to emphasize experimental physical phenomena and to present a modeling framework for two kinds of numerical simulations.

An experimental set-up was constructed to investigate the partial rub of a rotor due to contact with a non-rotating obstacle. The dedicated experimental bench as proposed is believed to be capable to simulate the rubbing phenomena which may occur in real rotating machinery. For highly-unbalanced rotating machines (generating "heavy" contact), some new characteristics are pointed out, primarily with regard to the orbit shape of the rotor. By examining more deeply the various phases of the measured orbit, one can say that this discrepancy could be caused by a very marked local deformation of the casing. This kind of casing "resonance" could be induced by the great flexibility of the casing or the bad attachment of the stator to its supporting structure. Recent investigations in the experimental set-up show that the stator is rather badly attached to its supporting structure than excessively flexible. The use of an "advanced" fully flexible diaphragm which incorporates the main features of the true structure (particularly realistic attachment conditions) seems to be mandatory for an accurate prediction of the bearing loads.

The suggested numerical approaches rely on finite element modeling for both the shaft and the stator. Firstly, the Harmonic Balance Method (HBM) has been presented which allows for fast assessment of the global nonlinear behavior of the system. This method solves the equation of motion in the frequency domain. The movement is thus supposed as periodic. On the other hand a time integration approach has also been presented which allows the transient effects to be taken into account. The combination of these two methods is believed to allow restricting the use of long transient simulations to a minimum.

Further investigations are required in the next future in order to analyze deeply different conditions other than stable light to heavy intermittent rubs, such as continuous full annular rub and even instable backward whirling motions. Moreover, an issue of a bad attachment of the stator to its structure will also be addressed. Simulations of more complex transient rubbing will then be performed and corresponding results will be confronted to experimental ones and reported in a forthcoming paper.

Figure 1 EDF

 1 Figure 1 EDF R&D EURoPE test rig

Figure 6

 6 Figure 6 measured angular velocity (left) and deceleration (right)

Figure 7

 7 Figure 7 model of the EURoPE test rig

Acknowledgements

The authors express their gratitude to the researchers and the engineers of EDF R&D AMA Department. This work has been also partially supported by the French National Agency (ANR) in the frame of its Technological Research COSINUS program. (IRINA, project ANR 09 COSI 008 01 IRINA).

Similarly to the previous case, the triggering of the contact is located in the upper part of the inner stator before the shaft crosses its critical speed. Image 2 shows severe intermittent rub conditions are experienced, contacts with rebounds occur (three contact points per revolution).

Under severe rub conditions, worth mentioning is the fact that the orbit is significantly enlarged and that it presents a more irregular shape during rub. In fact, one observes an almost triangular strongly distorted trajectory at the contact location during a few rotor revolutions. This behavior is due to sub-harmonic components and rub effects which contribute significantly, besides the 1x component, to compose the orbit. Some revolutions later (Image 4), there is no more than 2 contacts per revolution.

In this case, the rotor/stator coupling induces an augmentation of the shaft critical speed of about 70rpm. The contact forces reach values up to 1500N during the shut down.


M Dakel

S Baguet

R Dufour

Dynamic analysis of a harmonically excited on-board rotor-bearing system

The aim of this paper is to investigate the dynamic behavior of an on-board rotor mounted on elastic bearings in the presence of rigid support movements. The proposed on-board rotor model is based on the Timoshenko beam finite elements. It takes into account the six deterministic translations and rotations of its rigid support and the geometric asymmetry of shaft and/or rigid disks. Thus the obtained linear equations of motion of the rotating rotor in bending contain time-varying parametric terms which can lead to lateral dynamic instability. The influence of rotational or translational motions of the support is analyzed by means of orbits of the rotor, responses in the time domain and fast Fourier transforms (FFTs).

INTRODUCTION

Many industrial applications include rotating machines and the rotor plays a paramount role in them. There are many studies concerning the prediction of dynamics of rotor systems mounted on elastic bearings in the case of a fixed support [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF][START_REF] Genta | Dynamics of Rotating Systems[END_REF]. Some studies observed the instability of systems subjected to parametric excitations [START_REF] Dufour | Parametric instability of a beam due to axial excitations and to boundary conditions[END_REF]. Kang et al. [START_REF] Kang | Investigation on the steady-state responses of asymmetric rotors[END_REF] employed the Timoshenko beam finite elements for modeling asymmetric rotor-bearing systems. Some other works concentrated on the behavior of a rotor under seismic excitations [START_REF] Srinivasan | Seismic analysis of a rotor-bearing system[END_REF][START_REF] Hori | Earthquake-induced instability of a rotor supported by oil film bearings[END_REF][START_REF] Suarez | Seismic response of rotating machines[END_REF]. Subbiah et al. [START_REF] Subbiah | Response of rotors subjected to random support excitations[END_REF] studied the response of rotor systems under random support excitations using modal analysis methods. Lee et al. [START_REF] Lee | A finite element transient response analysis method of a rotor-bearing system to base shock excitations using the state-space Newmark scheme and comparisons with experiments[END_REF] focused on the experimental behavior of a rotor under shock support excitation. Da Silva Tuckmantel et al. [START_REF] Silva Tuckmantel | An analysis on the supporting structure representative model in rotating systems[END_REF] represented the supporting structure (foundation) of a rotating system by coupled as well as uncoupled modes for calculating the system response. In [START_REF] Feng | Rotor-model-based identification of foundations in rotating machinery using modal parameters[END_REF], experimental tests have been shown for a flexibly supported undamped rigid block foundation in rotating machinery. Duchemin et al. [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF] observed the stability of a simple rotor model under a sinusoidal support rotation. Driot et al. [START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF] described the orbits of a rotor induced by a harmonic rotational support movement. El-Saeidy and Sticher [START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF] obtained the responses of a rigid rotor-bearing system subjected to rotating mass unbalance plus harmonic support excitations. Das et al. [START_REF] Das | Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion[END_REF] investigated the active vibration control of a flexible rotor system excited by mass unbalance and periodic rotational motion of the support. Among all the literature mentioned above, there are references studying support-excited rotor systems and whose few works dealt with the harmonically excited on-board rotors [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF][START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF][START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF][START_REF] Das | Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion[END_REF]. Moreover, these references concentrated on the investigation of dynamic behavior of either simple rotors, rotors supported by elastic bearings with constant damping and stiffness coefficients or rotors excited by simple support motions. As a consequence, the applications proposed in these works are not suitable for realistic ones. In this paper, an improved model is presented. Namely, an asymmetric rotor is discretized using the finite element method based on the Timoshenko beam theory, mounted on hydrodynamic bearings linearized with damping and stiffness coefficients calculated using the Reynolds equation [START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF], and excited by different motions of its support. The rotary inertia, the gyroscopic inertia, the shear deformation of shaft and the geometric asymmetry of the rotor are taken into consideration. The effect of rotor internal damping is not included. The equations of motion point out periodic parametric terms due to the asymmetry of the rotor and time-varying parametric terms due to the rotational support excitations. In the presented application, the rotor mounted on elastic linear bearings is subjected to rotating mass unbalance combined with sinusoidal rotational or translational support motions. Numerical solutions are computed and analyzed by means of rotor orbits, time history responses and fast Fourier transforms (FFTs).

ANALYTICAL MODELING OF AN ON-BOARD ROTOR-BEARING SYSTEM

2.1

Preliminary calculations Three principal reference frames shown in Figure 1a are introduced to take into account the mobility of the rigid support of the rotor. They are attached to the ground R g , the rigid support R and the moving rotor R l . The rotational motions of the rotor support are defined by the angular velocity vector components ω x (t), ω y (t) and ω z (t) of the rigid support R with respect to the ground R g projected in the frame R. The translational motions of the rotor support are defined by the coordinates xO(t), yO(t) and zO(t) of the position vector O g O expressed in the frame R. The Euler angles Ψ(y,t), θ(y,t) and Φ(t) (Figure 1b) allow defining the orientation of the rotor R l with respect to its support R. The angular velocity vector of the rotor R l with respect to the ground R g is defined by:

   , , g g l l l l l l T R R R x y z R R R R ω ω ω ω ω ω ( 1
)
where T is a matrix transpose. The components (

Disk

Since the disk is considered to be rigid, only its kinetic energy Td is calculated as follows [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]: where md is the mass of the disk, of the disk center are functions of (ω x ,ω y ,ω z ) and (xO,yO,zO). The final expression of the kinetic energy of the disk having a mass center placed at the arbitrary abscissa yd relative to the frame R is given by [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]:

             2 1 with diag 2 2 g g g l l l d d d d d d d T R R R mo di y mo di d d m m m m m m m O R R m T I I I I I v ω I ω I (2)
               2 2 2 2 2 2 1 2 2 l l l l l l l l d d d mo x z y y di x z d d m m m O O O m T u v w I ω ω I ω I ω ω (3)

Shaft

Since the shaft is supposed to be flexible, it is characterized by the kinetic and strain energies and modeled by beam elements. The kinetic energy of a shaft can be obtained by taking an elementary volume of the shaft which can be considered as a disk of thickness dy. Thus the expression of the kinetic energy of a shaft has the following form [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]:

                          2 2 2 2 2 2 0 0 0 0 1 2 2 2 sh sh sh sh l l l l l l l l sh sh sh l l l l mo x z mo y di x z sh sh sh sh S sh S sh S O O O ρ S T u v w dy ρ I ω ω dy ρ I ω dy ρ I ω ω dy (4)
where ρsh, Ssh and lsh are respectively the density, the cross-section and the length of the shaft. sh mo S I and sh di S I are respectively the mean inertia of the cross-section and the inertia characterizing the asymmetry of the shaft. The rigid support motion relative to the ground has no influence on the strain energy of the shaft because the stresses depend only on the transverse deflection of the shaft with respect to the rotor support R. In addition to the bending deformation, the shear effects highlighted by Timoshenko and the centrifugal stress due to the support rotations are taken into account. The strain energy of a shaft is: [START_REF] Srinivasan | Seismic analysis of a rotor-bearing system[END_REF] where Esh and Gsh are respectively the Young's modulus and the shear modulus of the shaft (Gsh=Esh/(2(1+νsh)) hence νsh is the Poisson's ratio).

                                                                                                           2 2 2 2 2 2 0 0 0 2 2 1 2 2
                                                                                    0 0 2 2 2 2
mo sh k and di sh k are respectively the mean shear coefficient and the shear coefficient relative to the section asymmetry of the shaft.

Mass unbalance

Let us consider a concentrated mass unbalance mmu positioned at a point Pmu of the disk (ymu=yd) with a distance rmu from the geometric center of the shaft. Its initial angle with the Oz axis of the frame R at rest is ηmu. The components of the mass unbalance translational velocity vector are functions of (ω x ,ω y ,ω z ) and (xO,yO,zO) and are used in the kinetic energy which characterizes the mass unbalance [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF]: In the present study, the short bearing theory is considered (lbe/dbe≤1/8 where dbe=2rbe) and the static solution can be obtained using the following formulations deduced from the Reynolds equation with the Gümbel boundary conditions [START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF]:

   2 2 g mu R mu mu P m T v (6
                              1 2 2 2 2 2 2 2 2 2 16 1 1 Ω
and tan 4 1 be be be be be be r be be be be be be be

ε ε π ε ε r l π W μr l φ c d ε ε (7)
where εbe is the relative eccentricity (εbe=ebe/cbe) and μ is the fluid film dynamic viscosity. The above nonlinear equation is solved by an iterative Newton-Raphson method and provides the eccentricity εbe and then the static radial displacement ebe of the shaft center. The vector δbe,sep is obtained by a classical change of basis. The hydrodynamic fluid forces Fbe=<Fbe u ,Fbe w >R T produced by the bearings and expressed in the frame R can be obtained by integration of the fluid film pressure (Reynolds equation in the dynamic regime) over the bearing. In order to apply the Lagrange's equations, the virtual work δWbe of these forces has to be established:

   & , T be be be be be δW δ F δ δ δ (8)
If the lateral dynamic displacements δbe=<ube,wbe>R T of the shaft elastic line are assumed to be small in the vicinity of the static position δbe,sep, the linear analysis can be used by finding the first order Taylor series of the dynamic fluid film forces:

       & & ,
, , be be be be be sep be be be be

Equations of motion

The finite element method is chosen for discretizing the rotor and describing its flexural motion as a function of the nodal displacement vector defined by δ n =<u n ,w n ,θ n ,ψ n >R T , i.e., the rotor has two translations and two rotations at each node. The finite element used for the shaft modeling has two nodes and the shape functions are based on the Timoshenko beam theory. The effect of rotor internal damping is neglected. The linear equations of motion of the finite element rotor model are obtained after applying the Lagrange's equations to the energies for the disk, the shaft finite element and the mass unbalance as well as to the virtual work of the hydrodynamic bearings and assembling appropriately the produced vectors and matrices. They are written with respect to the noninertial reference frame connected to the rigid support R of the rotor:

               && & , , , r r r r r r r be r sep be r sep t t t t M δ C δ K δ F F δ 0 K δ (10)                                 , , , , , , , , , , , , , , ,Ω , , , , , , , , , , cos 2Ω sin 2Ω
M M M M C C C C C C K K K K K K K K K                 & & 2 2 2 2 2 2 2 2 , 2 , , 2 , , , , , , , , , , , ,Ω , , , 2 , , 2 , , 2 , , , , , , , , , , , , , , Ω x
ω ω ω ω ω ω ω ω ω ω ω ω K K K K K K K K K K          , , ,Ω , , , 2 , , , , , , , , 2 , , , , , , cos 2Ω
                                        , , (11)
F F F F F F F V V V V V V V V V                                    , Ω , , , , 2 , 2 , 2 , , , , , 2 2 ,
O O O O ω ω ω ω ω ω ω t x z ω y ω z ω ω ω y ω ω ω x ω ω z y ω x ω y ω ω ω x V V V V V V V V                                                , , , , , , , , , , , , , , , , , , , , Ω Ω
ω ω ω z ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω t ω ω ω ω t V V V V V V V V              & & , , , , , , 2Ω sin 2Ω 2Ω sin 2Ω ψ di x z y z θ di z x x y d sh su d sh su ω ω ω ω t ω ω ω ω t V V (12)
where && r δ , & r δ and r δ are respectively the acceleration, velocity and displacement vectors of the on-board rotor-bearing system of dimension 4(nesh+1)×1 hence nesh is the number of shaft finite elements. Mr(t), Cr(t) and Kr(t) are respectively the parametric mass, damping and stiffness matrices due to the asymmetry of the rotor and to the rotational motions of its support. Fr(t) is the external force vector. Fbe(δr,sep,0) is the hydrodynamic force vector caused by the bearings in the static case. This vector is opposite and equal to the rotor weight vector , r W d sh F . δr,sep is the static solution vector of the rotor due to the hydrodynamic bearing. The subscripts "d", "sh", "be" and "su" refer respectively to the disk, shaft, bearing as well as support and express the contribution to the phenomenon represented by the corresponding matrix. The superscript "g" refers to the gyroscopic effect, "e" signifies the shaft elasticity corresponding to the bending and shear deformations, "re" is the rotational effects due to the support rotations (these effects come from the kinetic energies of the disk and the shaft) and "gse" is the geometric stiffening effects corresponding to the centrifugal stress due to the support rotations. Vmu, Vd,sh,su and Vmu,su are load vectors associated respectively with the mass unbalance, the inertia force due to support motions and that due to coupling between both phenomena. The superscripts u, w, ψ and θ denote the direction of the action force components associated with the motions of the rotor support. The transient dynamic motion of the rotor is then obtained by solving Equation 10 by means of the implicit Newmark scheme based on the average acceleration. The static equilibrium position δr,sep is used to initialize the transient dynamic problem. The final integration time is chosen such that the transient effects have disappeared and the steady-state regime has been reached.

NUMERICAL EXAMPLES AND DISCUSSION

The symmetric rotor-linearized hydrodynamic bearing system presented in Figure 3 is subjected to rotating mass unbalance combined with sinusoidal rotational or translational support motions: the rotation around the Ox axis is given by ω x =ω x,a cos(Ω x t) in rad/s, while the translation along the Oz axis is expressed as zO=ZOcos(Ω z t) in m. The physical properties as well as the geometry of the rotor and the bearings are listed in Table 1. The shaft is discretized into eight identical 2-node Timoshenko beam finite elements. The disk is located at node 5 and bearings # 1 and # 2 are respectively placed at nodes 1 and 9. The rotor runs at a speed of rotation Ω=1200 rpm (=20 Hz=mass unbalance frequency) and its first four natural frequencies are 33.57 Hz, 52. All the presented orbits are dimensionless with respect to the bearing clearance cbe and are thus plotted in a unit circle. These orbits show the transient motion within the bearings from the static position Csep to the steady-state solution. Figure 4a presents a general view of the rotor orbit at the hydrodynamic bearings due to the mass unbalance effect. Since the disk is positioned at the middle of the shaft, the orbits at both bearings are identical in the case of a fixed support. The nonsymmetric damping and stiffness coefficients make the bearings anisotropic and the orbit elliptical with diagonal axes defining the phase between the mass unbalance excitation and the rotor response (Figure 4b). Even though the disk location is symmetric with respect to the bearings, the orbits are not identical in the case of a harmonic support rotation because of its normal and tangential accelerations. Focus is put on bearing # 2 which is more affected by the support rotation. The orbit characteristics (shape and magnitude) change with the amplitude (zO or ω x,a ) and frequency (Ω z or Ω x ) of the harmonic support excitation. As expected, in the case of very small amplitudes and frequencies of the support motions equal to the mass unbalance frequency, the orbits are similar to those produced when the support is fixed. Increasing the frequencies makes the orbit shapes more complicated; see Figures 5a and6a. The dynamic behavior is all the more modified as the frequency of the support excitation is high compared with the natural frequencies or the speed of rotation of the rotor. On the other hand, the amplitudes of the support motion change the x and z orbit magnitudes. Even if the support rotations have an influence on the mass unbalance excitation (see Equation 12), this influence does not appear clearly in the FFTs of the rotor motion during the harmonic support rotation because its amplitude is very small compared to the speed of rotation of the rotor (mass unbalance frequency).

                     

CONCLUSIONS

A finite element model is presented to analyze the dynamic behavior of a symmetric on-board rotor whose support is subjected to sinusoidal rotation or translation. The rotational effects and the geometric stiffening effects relative to the centrifugal stressing due to the support rotations are taken into account. The support rotations create time-varying parametric coefficients which can lead to lateral dynamic instability. It is noted that the shape and the magnitude of the rotor orbits can be significantly affected by the frequency and amplitude of the support motion respectively. It is shown that the mass unbalance forces and also the corresponding responses can depend on the support rotation around a transverse axis (in the presence of considerable rotation compared to the speed of rotation of the rotor) but not on the support translation along a lateral direction. The two frequency components due to the mass unbalance excitation and to the harmonic support motions appear in the time history responses and the FFTs of the rotor oscillations. In the case of large orbits, the assumption of a linearized hydrodynamic bearing model is questionable. A nonlinear bearing model is to be

  formulated as a function of (ψ,θ,Φ) and their time derivative as well as (ω x ,ω y ,ω z ). The rotor is supposed to rotate at a constant speed Ω. So the spinning angle Φ is replaced by Ωt and its derivative Φ • by Ω (• denotes differentiation with respect to time t). Let us consider a generic point C 0 along the elastic line of the nondeformed shaft. Its coordinates in the frame R are (0,y,0). It is interesting to study the translational displacements u(y,t) and w(y,t) of the point C 0 due to bending expressed respectively with respect to the Ox and Oz axes of the frame R.

  to the mean inertia of the disk mass and those due to the inertia modeling the geometric asymmetry of the disk. of the disk mass about the Oy axis. The translational velocity vector components

  and kbe are the damping and stiffness matrices of the linearized hydrodynamic bearing (Figure2b) whose analytical expressions can be found in[START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF]. (a) (b) Figure 2. (a) Hydrodynamic bearing, (b) Bearing damping and stiffness coefficients 10th IMechE International Conference on Vibrations in Rotating Machinery (VIRM10), Sep 2012, London, UK, pp.C1326/024

  7 Hz, 542.4 Hz and 1647.2 Hz. The relative static equilibrium position of the geometric center of the shaft in the fluid film bearings is given by δbe,sep/cbe=<-0.2855,-0.8789>R T . The bearing damping and stiffness matrices are expressed in what follows:

Figure 3 .

 3 Figure 3. Schematic diagram of the investigated rotor-bearing system Table 1. Main characteristics of the investigated rotor and the bearings Density of both disk and shaft material Radius, thickness and location of the disk shaft material Radius and length of the shaft Mass unbalance

Figure 4 .

 4 Figures5 and 6display the rotor orbits at bearing # 2 during the harmonic translational or rotational motions of the support. Even though the disk location is symmetric with respect to the bearings, the orbits are not identical in the case of a harmonic support rotation because of its normal and tangential accelerations. Focus is put on bearing # 2 which is more affected by the support rotation. The orbit characteristics (shape and magnitude) change with the amplitude (zO or ω x,a ) and frequency (Ω z or Ω x ) of the harmonic support excitation. As expected, in the case of very small amplitudes and frequencies of the support motions equal to the mass unbalance frequency, the orbits are similar to those produced when the support is fixed. Increasing the frequencies makes the orbit shapes more complicated; see Figures5a and 6a. The dynamic behavior is all the more modified as the frequency of the support excitation is high compared with the natural frequencies or the speed of rotation of the rotor. On the other hand, the amplitudes of the support motion change the x and z orbit magnitudes.

Figure 5 .Figure 6 .

 56 Figure 5. Relative orbits of the rotor at bearing # 2 for two amplitudes of sinusoidal support translations: (a) ZO=10 -5 m, (b) ZO=5×10 -5 m

Figure 7 .Figure 8 .

 78 Figure 7. Relative responses and FFTs of the lateral vibrations at bearing # 2 for support translations of amplitude ZO=5×10 -5 m and frequency: (a) Ω z =20 Hz, (b) Ω z =50 Hz, (c) Ω z =80 Hz Displacement in the Ox direction Displacement in the Oz direction

  [START_REF] Genta | Dynamics of Rotating Systems[END_REF]

	U	sh	sh mo l sh S E I	sh	ψ y	θ y	dy	mo sh sh G k S sh	sh l	u y	ψ	w y	θ	dy	sh di sh S E I	sh l	ψ y	θ y	dy
			di sh sh sh G k S	u y	ψ	w y	θ

  BearingFigure2ashows a simple diagram of a hydrodynamic bearing which is composed of a fixed journal containing a rotating shaft. The points O and O l =Csep represent respectively the bearing center and the shaft center. The radius, length and clearance of the bearing are respectively rbe, lbe and cbe=rbe-rsh where rsh is the shaft radius. At a constant rotor speed Ω and for a constant static load Wr created by the rotor weight, the shaft center Csep in the bearing holds a static equilibrium position defined by the vector δbe,sep=<ube,sep,wbe,sep>R T expressed in the frame R or

	equivalently by the eccentricity	be e		δ	, be sep	of the shaft center in the journal and

)

10th IMechE International Conference on Vibrations in Rotating Machinery (VIRM10), Sep 2012, London, UK, pp.C1326/024

2.2.4

the attitude angle φbe between the Wr load direction and the line of centers OCsep.

& & & 2 2

& & 2

& & 2 2

& & 2 2 2

& &

& & & &

& & & & 2 2 2

& & & & & & 2 2

10th IMechE International Conference on Vibrations in Rotating Machinery (VIRM10), Sep 2012, London, UK, pp.C1326/024


M Dakel

S Baguet

R Dufour
email: regis.dufour@insa-lyon.fr

STEADY-STATE RESPO SE OF A ROTOR EXCITED BY COMBI ED ROTATIO AL A D TRA SLATIO AL BASE MOTIO S

Keywords: Dynamics of rotors, on-board rotor dynamics, base excitation, analytical solution

In the transportation domain, the on-board rotor in bending is subjected not only to rotating mass unbalance but also to support movements. The equations of motion in bending of the rotating rotor take into account the geometric asymmetry of disks and/or shaft and consider six types of deterministic support motions. The application of Lagrange's equations using the finite element method based on the theory of Timoshenko leads to the equations of motion which highlight periodic parametric terms due to the asymmetry of the rotor and time-varying parametric terms due to the rotational base excitations. When the rotor base is subjected to combined rotation and sinusoidal translation, analytical solutions are derived and analyzed by means of Campbell diagrams and steady-state responses.

Introduction

Rotordynamics with a fixed support has been extensively predicted [START_REF] Rao | Rotordynamics[END_REF][START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF][START_REF] Genta | Dynamics of Rotating Systems[END_REF]. Other works concentrate on the behavior of a rotor in the seismic excitation cases [START_REF] Hori | Earthquake-induced instability of a rotor supported by oil film bearings[END_REF][START_REF] Suarez | Seismic response of rotating machines[END_REF][START_REF] Bachelet | Rotors under seismic excitation: A spectral approach[END_REF]. The effect of the flexible foundation on the rotor dynamics has been studied using a finite element modeling [START_REF] Kang | An investigation in stiffness effects on dynamics of rotor-bearing-Foundation systems[END_REF] and an experimental investigation of a rotor [START_REF] Edwards | Experimental identification of excitation and support parameters of a flexible rotor-bearings-foundation system from a Single Run-Down[END_REF]. In [START_REF] Gaganis | Modal analysis of rotor on piecewise linear journal bearings under seismic excitation[END_REF], the nonlinear bearing coefficients are considered to be piecewise constant using the El Centro earthquake data. Certain study focuses on the experimental behavior of a rotor under shock base excitation [START_REF] Lee | A finite element transient response analysis method of a rotor-bearing system to base shock excitations using the state-space Newmark scheme and comparisons with experiments[END_REF]. When the rotor support is induced by a deterministic harmonic motion, the response to this excitation [START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF], the stability chart [START_REF] Duchemin | Behavior and stability of a rotor under base excitation[END_REF], the comparisons between experimental and numerical orbits [START_REF] Driot | Theoretical and experimental analysis of a base excited rotor[END_REF] as well as the active vibration control of the rotor [START_REF] Das | Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion[END_REF] are studied and presented. In this paper, we present a symmetrical rotating rotor model whose base is subjected to a constant angular velocity and a periodic translation. Campbell diagrams, mass unbalance responses and responses to translational base excitations are plotted as a function of the base rotations which can drastically change the behavior of the rotor.

Dynamical behavior modeling

The rotor support is supposed to be sufficiently rigid compared to the flexible shaft of the rotor. Its motion is described by three rotations and three translations. The investigated rotor and the coordinate system attached to the support are presented in Fig. 1. The derivation of the equations of motion has been established in [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF][START_REF] Lee | A finite element transient response analysis method of a rotor-bearing system to base shock excitations using the state-space Newmark scheme and comparisons with experiments[END_REF][START_REF] Duchemin | Behavior and stability of a rotor under base excitation[END_REF] and is briefly described as follows:

• The disc is characterized by its kinetic energy because it is considered rigid.

• The elementary kinetic energy of an elementary shaft volume is considered as a generalization of the disc case. Moreover, the strain energy of a shaft finite element is calculated using the Timoshenko beam theory. • The mass unbalance located on the disc is characterized by its kinetic energy. The nodal displacement vector is defined by q n =<u n ,w n ,θ n ,ψ n > T . The equations of motion of the finite element rotor model are obtained after applying the Lagrange's equations to the energies for the disc, the shaft finite element as well as the mass unbalance and assembling the corresponding matrices:

( ) ( ) ( ) ( ) t t t t + + = r r r r M q C q K q F ɺɺ ɺ (1)
where M r (t), C r (t) and K r (t) are the parametric matrices of mass, damping and stiffness of an asymmetrical rotating rotor on moving base. F r (t) is the total force vector. Now, the shaft and the disc are assumed to be symmetric and the base excitations are assumed to consist of a constant angular velocity ω x (or ω y ) around the Ox axis (or the Oy axis) and a periodic translation along the Oz axis given by z O =Z O sin(Ω z t). The equations of motion (1) in this case include constant terms and may be rewritten in one of the two following forms (Eq. ( 2) corresponds to ω x and Eq. ( 3) to ω y ):

y d l s e d r s r d O y x z θ n ψ n u n w n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 cos sin sin x x x z x z O t t Z t + Ω + + + ω = Ω + ω Ω + Ω + ω Ω + Ω + ω Ω gse g re c c u b b1 u,b1 s s s u b u,b1 Mq C q K K K q V V V V V ɺɺ ɺ (2) ( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
+ Ω + ω + + Ωω + ω = Ω + Ωω + ω Ω + Ω + Ωω + ω Ω - Ω ω Ω + Ω + ω Ω g re re c b u b2 b3 c s s u u,b2 u,b2 c s b b Mq C C q K K K q V V V V V V ɺɺ ɺ ( 3
)
where Ω is the speed of rotation of the rotor. V u , V b and V u,b are load vectors associated respectively with the mass unbalance, the force inertia due to base motions and that due to coupling between both phenomena. The superscript "g" refers to the gyroscopic effect, "re" is the rotational effects due to the base rotation and "gse" is the geometric stiffening effects corresponding to the centrifugal stress due to the base rotation. In our case, the system of Eq. ( 1) contains constant matrices (according to Eqs. ( 2) and ( 3)) and is transformed into a first-order system:

( ) t           = +         - -             -1 -1 -1 r r r r r r 0 0 I q q q q M F M K M C ɺ ɺɺ ɺ (4)
The eigenvalues of the above matrix is calculated for finding the natural frequencies f of the rotor. Here only the steady-state responses are calculated. The general solution of the equations of motion (4) is not considered and its particular solution can be expressed by:

( ) ( ) ( ) ( ) cos sin cos sin z z t t t t = Ω + Ω + Ω + Ω c s c s u u b b q q q q q (5)
where c u q , s u q , c b q and s b q are unknown displacement amplitude vectors. Use q to obtain the velocity and acceleration vectors. Substituting them in Eq. ( 4) gives:

2 2 2 2 z z z z     +   - Ω Ω       =           -Ω - Ω +             - Ω Ω       =           -Ω - Ω      
F F q K M C C K M q F F q F K M C C K M q F (6)

Results and Discussion

The material properties and the geometry of the rotor are given in Table 1.

Table 1. Main characteristics of the investigated rotor. Figure 2 shows the Campbell diagrams of the rotor subject either to ω x or ω y angular velocity. The base rotation ω x breaks the symmetry of the rotor behavior; see its Campbell diagram presented in Fig. 2(a). The rotational effects due to ω x decrease the natural frequencies while the centrifugal stiffening relative to ω x increases them especially at the higher speeds of rotation of the rotor. Unlike the previous case, the solicitation due to the base rotation ω y does not bring any asymmetry to the rotor, but shifts the natural frequencies; see its Campbell diagram presented in Fig. 2 ), but three in the presence of ω x (see Fig. 3(a)). These responses confirm that the rotor keeps a symmetrical behavior in the case of the base rotation around the Oy axis, but has a non-symmetrical behavior in the case of the base rotation around the Ox axis. We note that the position of the critical speed is shifted towards the lower values of the speed of rotation of the rotor when the angular velocity ω y increases. Figure 4 presents the evolution of the frequency responses at the middle of the rotor running at a speed of rotation of 2000 rpm and subject to a harmonic base translation of amplitude 2 × 10 -4 m in the z direction. In the presence of ω x or ω y angular velocity, the responses exhibit always three critical frequencies (three peaks) corresponding to the rotor natural frequencies at 2000 rpm. In the case of ω y , the harmonic translation along the Oz axis produces a very small peak compared to the other two peaks (see Fig. 4(b)). This is because Eq. (3) presents a negative cosine term containing the frequency Ω z of the base translation.

Conclusions

A finite element model is presented for analyzing the dynamical behavior of a symmetrical on-board rotor whose support is subjected to combined constant rotation and sinusoidal translation. The rotational effects and the geometric stiffening effects relative to the centrifugal stressing due to the support rotation are taken into account. It is shown that support rotation not only changes the natural frequencies of the rotor but can also break the symmetry of the dynamical behavior in the case of the rotation around a transverse axis. The sinusoidal support translation does not change the rotor natural frequencies. It has only an influence on the rotor responses and the shape of the orbits.

Fig. 1 .

 1 Fig. 1. Description of the investigated rotor.

Density of both disc and shaft

  Figure2shows the Campbell diagrams of the rotor subject either to ω x or ω y angular velocity. The base rotation ω x breaks the symmetry of the rotor behavior; see its Campbell diagram presented in Fig.2(a). The rotational effects due to ω x decrease the natural frequencies while the centrifugal stiffening relative to ω x increases them especially at the higher speeds of rotation of the rotor. Unlike the previous case, the solicitation due to the base rotation ω y does not bring any asymmetry to the rotor, but shifts the natural frequencies; see its Campbell diagram presented in Fig.2(b).

Fig. 2 .

 2 Fig. 2. Campbell diagrams for the first four natural frequencies of the rotor during the support rotation: (a) with ω x =0 (solid line), 4π rad/s (2 Hz, dotted line), 10π rad/s (5 Hz, dashed line), 20π rad/s (10 Hz, dashed-dotted line); or (b) with ω y =0, 4π, 10π, 20π rad/s.

Figure 3

 3 Figure3represents the mass unbalance responses at the middle of the shaft subject either to ω x or ω y angular velocity. The rotor has only one critical speed in the presence of ω y (see Fig.3(b)), but three in the presence of ω x (see Fig.3(a)). These responses confirm that the rotor keeps a symmetrical behavior in the case of the base rotation around the Oy axis, but has a non-symmetrical behavior in the case of the base rotation around the Ox axis. We note that the position of the critical speed is shifted towards the lower values of the speed of rotation of the rotor when the angular velocity ω y increases.

Fig. 3 .

 3 Fig. 3. Mass unbalance responses during the support rotation around: (a) the Ox axis; or (b) the Oy axis.

Fig. 4 .

 4 Fig. 4. Dynamical responses to the periodic base translation at 2000 rpm speed of rotation of the rotor during the base rotation around: (a) the Ox axis; or (b) the Oy axis.


Information Leakage in a Music Score

François Fages

Inria Saclay-Île de France, Palaiseau, France When I was invited to contribute to a Festschrift in honor of Catuscia Palamidessi, I immediately thought I should not miss the opportunity to express my friendly feelings and admiration for the research career of a good friend of 30 years.

Initially, we had some common research axes on Logic Programming and Concurrent Constraint Programming. The GULP conference on these topics in Italy was a high place where Catuscia had no equal to defend, between Pisa and Florence, what should be the true Italian pronunciation, but with too subtle arguments for a foreigner. In 1990, the ICLP conference in Jerusalem and Eilat was a memorable mark of Catuscia's passion for Concurent Logic Programming. In 2000, our small CSE Penn State-Inria workshop on "Concurrency and Logic" was premonitory of a new life for Catuscia at Inria. In the last decade however, our research interests evolved in different directions, respectively on quantitative information flows and quantitative biology.

When reading the recent papers of Catuscia on information leakage in information flows, in the quest of finding something to write related to those topics, a music unexpectedly came to my mind, with two concurrent voices playing obsessively on some jazzy rhythm, responding each other, and producing some bits of information about a secret message subject to interpretation, as always in music.

It is my pleasure to dedicate that little piece of music, called "Concur in C", to Catuscia for her 60th birthday. The music score is given below1 .

But could we try to apply Catuscia's theory on information leakage in information flows to the notes of each voice, in the hope of revealing something more on the meaning of this mysterious musical message remained enigmatic to the composer?

In information flow theory, a channel, or a voice in our musical interpretation, is a triple (X, Y, C), where X and Y are finite sets (of secret input values and produced output music notes) and the channel matrix C is an |X| × |Y | matrix giving the probability of getting output y when the input is x. Given a prior distribution π on X, the joint distribution p(x, y)

= π[x]C[x, y] on random variables X × Y is the (unique) joint distribution that gives marginal probabili- ties p(x) = y p(x, y) = π[x], conditional probabilities p(y|x) = p(x, y)/p(x) = C[x, y] (if p(x) is nonzero),

and similarly p(y) and p(x|y).

The marginal distribution of the music notes in a voice are given by the frequencies of the notes in that voice. In Concur in C, the upper and lower voices happen to play the same number of notes, 186, with slightly different distributions and slightly higher Shannon's entropy (-186 i=1 p i (y).log 2 (p i (y))), i.e. more information, in the upper voice:

number of C C# D D# E F F# G G# A A# B total occurrences
Db Eb Gb Ab Bb upper voice 32 0 0 54 2 14 20 40 0 0 22 2 186 p u (y) 0.17 0 0 0.29 0.01 0.07 0.11 0.22 0 0 0.12 0.01 entropy 2.56 lower voice 41 0 0 36 0 22 27 39 0 0 21 0 186 p l (y) 0.22 0 0 0.19 0 0.12 0.15 0.21 0 0 0.11 0 entropy 2.53

In music, it is expected that the upper voice contains more information than the lower voice usually dedicated to accompaniment. But here, the two voices play similar melodies in counterpoint which results in similar entropy values for both voices. It would be interesting to test this approach on more elaborate scores. Thank you Catuscia for having once again inspired me, in an unexpected fashion, this self-surprising gift with curious questions about the quantification of information in music.

There has always been something magic in you. Happy 60th Birthday Catuscia!

"Concur in C" can be heard with several orchestrations (piano, flute-saxophone, flute-organ-bass, big band) on http://lifeware.inria.fr/~fages/music/

Acknowledgments: special thanks go to Hubert Garavel for having improved the layout of my music score, and for his tolerance regarding my parallel fifths.