Mzaki Zaki Dakel
email: mzaki.dakel@insa-lyon.fr

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

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

Bifurcation

Bifurcation Analysis of a Non-linear On-Board Rotor-Bearing System

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INTRODUCTION

The rotating machines occupy an important position in our life and play a paramount role in the modern industry. The rotor represents their key component and is subject to base excitations when it is embedded in moving systems. Vehicle turbochargers and gas turbine rotors installed in power plants are especially concerned with such a situation.

The rotor systems are complex and their dynamic analysis is rather difficult. Several studies focused on making a clear understanding of the dynamics of symmetric/asymmetric rigid/flexible rotor systems supported by linear/non-linear elastic bearings in the case of a fixed base, see, for example, Lalanne and Ferraris [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF] and Genta [START_REF] Genta | Dynamics of Rotating Systems[END_REF]. Berlioz et al. [START_REF] Berlioz | Bifurcation in a Nonlinear Autoparametric System Using Experimental and Numerical Investigations[END_REF] investigated numerically and experimentally the bifurcation of solutions of the non-linear equations of motion with periodic parameters for an on-board pendulum. Oncescu et al. [START_REF] Oncescu | Investigation of the Stability and Steady State Response of Asymmetric Rotors, Using Finite Element Formulation[END_REF] generalized a finite element procedure for rotor-bearing systems to include the effects of the asymmetry of shaft and/or bearing and predicted the dynamic behavior of such systems.

The rotor-hydrodynamic bearing system is fully non-linear due to non-linear fluid forces acting locally on it. Li and Xu [START_REF] Li | A Method to Determine the Periodic Solution of the Non-linear Dynamics System[END_REF] studied a Jeffcott rotor mounted on fluid film infinite-length bearings to predict periodic orbits, their periods and their stability by employing the generalized shooting method. Bifurcation diagrams as well as harmonic, sub-harmonic and quasi periodic responses and their Poincaré maps were investigated for a flexible rotor (Chen and Yau [START_REF] Chen | Bifurcation in a Flexible Rotor Supported by Short Journal Bearings With Nonlinear Suspension[END_REF]) and a vertical rigid rotor (Shi et al. [START_REF] Shi | Nonlinear Dynamic Analysis of a Vertical Rotor-Bearing System[END_REF]) supported by hydrodynamic short bearings.

Srinivasan and Soni [START_REF] Srinivasan | Seismic Analysis of a Rotor-Bearing System[END_REF] examined the influence of spin, base rotation as well as axial force and axial torque on the seismic response of a rotor-bearing system. Hori and Kato [START_REF] Hori | Earthquake-Induced Instability of a Rotor Supported by Oil Film Bearings[END_REF] studied the seismic response of a Jeffcott rotor supported by fluid film bearings to a real seismic wave and investigated its stability by calculating the locations of the disk and journal centers. Kang et al. [START_REF] Kang | An Investigation in Stiffness Effects on Dynamics of Rotor-Bearing-Foundation Systems[END_REF] and Cavalca et al. [START_REF] Cavalca | An Investigation on the Influence of the Supporting Structure on the Dynamics of the Rotor System[END_REF] observed the influence of the flexible supporting structure on the dynamic characteristics of the rotor-bearing systems.

The number of works dealing with the prediction of dynamic behavior of the rotor systems excited by a sinusoidal motion of their base remains actually low. Duchemin et al. [START_REF] Duchemin | Dynamic Behavior and Stability of a Rotor Under Base Excitation[END_REF] employed the method of multiple scales for a simple rotor model under a sinusoidal rotation of the base in order to observe its dynamic stability. They showed also experimental results to validate the analytical study. Driot et al. [START_REF] Driot | Theoretical and Experimental Analysis of a Base-Excited Rotor[END_REF] presented numerical orbits of a rotor induced by a sinusoidal rotational excitation of its base. According to their work, the comparison between numerical and experimental results was widely satisfactory. 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] calculated the responses of a rigid rotor-bearing system subject to rotating mass unbalance and sinusoidal movements of the base along or around lateral directions using analytical solutions in the case of linear bearings and a time integration scheme in the case of a bearing cubic non-linearity. Das et al. [START_REF] Das | Active Vibration Control of Unbalanced Flexible Rotor-Shaft Systems Parametrically Excited Due to Base Motion[END_REF] executed a numerical simulation of a flexible rotor system excited by a mass unbalance and a sinusoidal rotation of the base in order to investigate its active vibration control which was successful for avoiding the lateral parametric instability. Dakel et al. [START_REF] Dakel | Investigation on the Dynamics of an On-Board Rotor-Bearing System[END_REF] computed the orbits and their spectral analysis for a rotor mounted on linearized hydrodynamic short bearings and excited by combined rotational and translational motions of its base. They predicted also the linear dynamic behavior of symmetric and asymmetric rotors on rigid bearings subject to combined constant rotation and sinusoidal translation of the base [START_REF] Dakel | Steady-State Dynamic Behavior of an On-Board Rotor Under Combined Base Motions[END_REF]. More recently, they presented a comparison between linear and non-linear responses of a rotor mounted on hydrodynamic short bearings and excited by sinusoidal motions of the base to assess the validity range of the linearized bearing model [START_REF] Dakel | Nonlinear Dynamics of a Support-Excited Flexible Rotor With Hydrodynamic Journal Bearings[END_REF].

The present work is mainly based on the model presented in [START_REF] Dakel | Nonlinear Dynamics of a Support-Excited Flexible Rotor With Hydrodynamic Journal Bearings[END_REF] but investigates in more detail the non-linear effects due to the combination of a non-linear rotor-bearing system (because of fluid film forces) and base excitations. The proposed on-board rotor is discretized using the finite element method based on the Timoshenko beam theory and excited by combined deterministic motions (three rotations and three translations) of its rigid base. The rotary inertia, the gyroscopic inertia, the shear deformation of shaft as well as the geometric asymmetry of shaft and/or rigid disk are taken into account. The non-linear hydrodynamic fluid forces within the bearing are evaluated by the Reynolds equation and treated as restoring forces acting on the shaft. The application of the Lagrange's equations gives the non-linear differential equations of motion of the rotor in bending with respect to a non-inertial reference frame linked with the rigid base. These equations point out periodic parametric terms due to the geometric asymmetry of the rotor and time-varying parametric terms due to the rotational excitations of the base. The transient implicit Newmark time-step integration scheme combined with the Newton-Raphson iterative strategy is employed to treat the equations of motion. In addition, in the case when sinusoidal motions of the rotor base yield periodic (harmonic and subharmonic) responses, an optimized shooting method is used to treat the equations of motion. The non-linear phenomena identified in the on-board rotor are described by means of bifurcation diagrams, rotor orbits as well as Poincaré maps.

PRELIMINARY CALCULATIONS

The rotor frequently consists of the disk, shaft, mass unbalance, bearing and base. The excitations taken into account are due to the mass unbalance and to the rigid base motions. Three principal Cartesian coordinate systems shown in Fig. 1 are introduced to take into consideration the mobility of the rotor base. They are attached to the ground R g , the rigid base R and the moving rotor R l .

The translational motions of the rotor base are defined by the coordinates x O (t), y O (t) and z O (t) of the position vector O g O of the origin O expressed in the frame fixed to the base R. The rotational motions of the rotor base are defined by the angular velocity vector components ω x (t), ω y (t) and ω z (t) of the rigid base R with respect to the ground R g projected in the frame R.

A generic point C init on the elastic line of the non-deformed shaft is considered. Its coordinates in the frame R are (0,y,0). Its lateral dynamic displacements u(y,t) and w(y,t) due to bending are expressed respectively with respect to the Ox and Oz axes. Additional details on the kinematics of the on-board rotor can be found in [START_REF] Dakel | Steady-State Dynamic Behavior of an On-Board Rotor Under Combined Base Motions[END_REF][START_REF] Dakel | Nonlinear Dynamics of a Support-Excited Flexible Rotor With Hydrodynamic Journal Bearings[END_REF].

ENERGY AND VIRTUAL WORK CALCULATIONS

Disk, Shaft and Mass Unbalance. The necessary equations have been established in [START_REF] Dakel | Steady-State Dynamic Behavior of an On-Board Rotor Under Combined Base Motions[END_REF][START_REF] Dakel | Nonlinear Dynamics of a Support-Excited Flexible Rotor With Hydrodynamic Journal Bearings[END_REF] and the calculations are briefly described hereafter.

Since the disk is assumed to be rigid, only its kinetic energy is computed. The kinetic energy of a shaft elementary volume is considered as a generalization of the case of a disk. Moreover, the strain energy of the shaft is calculated using the Timoshenko beam theory. The mass unbalance located on the disk is characterized by its kinetic energy.

Hydrodynamic Journal Bearing. The short bearing theory is considered in this study. At a constant speed of rotation Ω of the rotor and for a constant static load W r created by the rotor weight, the shaft geometric center C be 0 in the bearing holds a static equilibrium position defined by the displacement vector δ be 0 =<u be 0 ,w be 0 > R T expressed in R. In the dynamic regime, the dynamic transverse displacement vector of the shaft center is δ be =<u be ,w be > R T . Assuming an isothermal and laminar flow of an isoviscous incompressible fluid, the hydrodynamic fluid pressure field is analytically calculated with the Reynolds equation. Its integration using the Gümbel boundary conditions leads to the dynamic polar (tangential and radial) components of the hydrodynamic force vector expressed in the polar frame linked with the hydrodynamic bearing R be and defined by the following formulations (see [START_REF] Frêne | Hydrodynamic Lubrication Bearings and Thrust Bearings[END_REF])

( ) ( ) ( ) ( ) ( ) 3 2 2 2 2 2 3 2 2 2 2 2 4 1 2 , 2 2 1 1 2 2 2 , 1 2
1
ta be be be be be be be be be be be be ra be be be be be be be be be

r l F c r l F c µ ε π ε ε ϕ ε πε ε µ ε ϕ ε ε   = + - Ω -     -   +   = - + Ω -   - -   ɺ ɺ ɺ ɺ (1) with 2 2
; tan , be be be be be be be ; , be be be be be be be be be be be be be be

u w u c w ε ϕ + = = (2)
u u w w u w w u c c ε ϕ ε ε + - = = ɺ ɺ ɺ ɺ ɺ ɺ (3)
where µ be , r be , l be and c be are respectively the dynamic viscosity, radius, length, clearance of the bearing. ε be and φ be are the relative dynamic eccentricity and the dynamic attitude angle.

The components of the dynamic fluid force vector

F be =<F be u ,F be w > R
T projected in the frame R are obtained by a classical change of basis between the frames R be and R. They are non-linear functions of the dynamic transverse displacement and velocity vectors ( ) , be be δ δ ɺ . Since the bearing contribution appears as forces F be external to the rotor, their virtual work δW be has to be established to apply the Lagrange's equations ( )

, .

T be be be be be

W δ δ = F δ δ δ ɺ (4)

EQUATIONS OF MOTION

The finite element method is selected to discretize the rotor and to describe 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 lateral translations and two rotations at each node. The shaft finite element has two nodes and the shape functions are based on the Timoshenko beam theory.

The non-linear equations of motion of the on-board rotorbearing system in bending are obtained after applying the Lagrange's equations to the energies for the disk, the shaft finite elements 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 non-inertial frame connected to the base R as follows

( ) ( ) ( ) ( ) ( ) , , be t t t t + + = + M δ C δ K δ F F δ δ ɺɺ ɺ ɺ (5)
with δ ɺɺ , δ ɺ and δ the acceleration, velocity and displacement vectors of the on-board rotor-bearing system of dimension 4(n esh +1)×1 where n esh is the number of shaft finite elements. M(t), C(t) and K(t) are respectively the mass, damping and stiffness matrices with time-varying coefficients due to the asymmetry of the rotating rotor and to its moving base. F(t) is the external linear force vector due to the mass unbalance and to the rotational and translational motions of the base. In addition, the external non-linear hydrodynamic force vector ( )

, be F δ δ ɺ of
the bearings is opposite to the rotor motion.

The matrices of Eq. ( 5) are defined as follows

( ) 2 2
, , , cos 2 sin 2 ,

c s d sh d sh d sh t t t = + Ω + Ω M M M M (6) ( ) 2 2 , , , , , , , , cos 2 sin 2
,

y g c g s g d sh d sh d sh re y d sh b t t t ω ω = Ω + Ω Ω + Ω Ω + C C C C C (7) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 , , , , , , 2 , , , , , , , , , 2 , , 2 , , , , , , , , , , , , , , , , cos 2
+ + + + + + K K K K K K K K K K K K K K ɺ ɺ ɺ ɺ ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 , , , 2 , , , , , , , , 2 2 , , , , , , , , , , , , 2 , , , , , , , , , , 2 , , , , cos 2 s
+ + Ω + + + K K K K K K K K K ɺ ɺ in 2 . t Ω (8)
The subscripts "d", "sh" and "b" refer to the disk, shaft as well as base respectively and express the contribution to the phenomenon represented by the corresponding matrix. The superscripts "c 2 " and "s 2 " signify the geometric asymmetry of the rotor expressed in terms of the time-varying trigonometric functions cos2Ωt and sin2Ωt. The superscript "g" stands for the rotor gyroscopic effect, "e" for the shaft elasticity corresponding to the bending and shear deformations, "re" for the rotational effects corresponding to the rotational motions of the base (these effects come from the kinetic energies of the disk and the shaft) and "gse" for the geometric stiffening effects associated with the centrifugal stress due to the rotational motions of the base.

The vector F(t) is expressed as follows

( ) ( ) ( ) ( ) ( ) ( ) 2 2 , , , , , , , , cos 2 sin 2
= + + + + Ω + Ω F F F F F F F (9) with , , , r r W W d sh d sh r W = - F V (10)
( ) x z

1 1 2 2 cos sin , c s mu mu mu t t t = Ω Ω + Ω Ω F V V (11) z g x g y g O g z x y O z l x l y l O l ( ) ( ) ( ) 2 2 1 1 1 1 2 1 1 2 1 1 1 2 2 1 1 1 , , , , 2 2 , , , , , , , 2 , , , , , 2 , , , , , , 2 2 ,
+ + Ω + + Ω + + + + Ω F V V V V V V V V V V V V ɺ ɺ ɺ ɺ , (12) ( ) ( ) ( ( ) ( )) ( ) ( ( ) ( )) ( ) ( ) ( ) ( ) , , , , 2 2 , , 2 2 , , , , , , , , 2 2 2 2
θ ψ ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω = - + - + + - - - + - + - + + - - - + - + + - - + - - - F V V V V V V V ɺ ɺɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺɺ ɺ ɺ ɺ ɺ ɺ ( ) ( ) , , , ,
,

y x x y y z y z sh b d sh b ψ θ ω ω ω ω ω ω Ω + + Ω + V (13) ( ) ( ) ( ) 2 2 2 , , , , , , , , 2 2
,

c c x z y z d sh b d sh b c z x x y d sh b t θ ψ ω ω ω ω ω ω ω ω = - -Ω - + + Ω + F V V ɺ ɺ (14) ( ) ( ) ( ) 2 2 2 , , , , , , , , 2 2 ,
s s x z y z d sh b d sh b s z x x y d sh b t ψ θ ω ω ω ω ω ω ω ω = -Ω - + + Ω + F V V ɺ ɺ (15)
where

V mu (F mu (t)), V d,sh,b (F d,sh,b (t)) and V mu,b (F mu,b (t)
) are load vectors (force vectors) associated respectively with the mass unbalance, the inertia force due to base motions and that due to coupling between both phenomena.

, r W d sh F is the rotor weight vector. The superscripts "c 1 " and "s 1 " denote the components of the mass unbalance force expressed in terms of the time-varying trigonometric functions cosΩt and sinΩt. The superscripts u, w, ψ and θ denote the direction of the action force components associated with the motions of the rotor base. As already stated above, the equations of motion are nonlinear. This is due to the hydrodynamic bearings (local components) even if they do not concern all the degrees of freedom of the rotor system. In the two following sections (i.e., Sections 5 and 6), the treatment of these equations is detailed.

TRANSIENT DYNAMIC ANALYSIS

The resolution of the complete non-linear dynamic problem requires solving simultaneously the equations of motion (i.e., Eq. ( 5)) of the rotor-bearing system and Eqs. ( 1)-( 3) calculating the non-linear hydrodynamic bearing forces governed by the Reynolds equation in the dynamic regime. Therefore the nonlinear analysis uses a complex procedure combining the Newton-Raphson incremental-iterative technique with the Newmark time integration algorithm. The semi-discrete equations of motion (Eq. ( 5)) applied at time t i+1 (1≤i≤n ts +1 where n ts is the number of time steps) can be given in the following residual form

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 , . i i i i i i i i be i i t t t t + + + + + + + + + + = + + - - = G δ M δ C δ K δ F F δ δ 0 ɺɺ ɺ ɺ (16)
The velocity and displacement vectors of the rotor nodes between t i and t i+1 =t i +∆t, ∆t being the time step, are approximated by the following general formulations of the implicit Newmark scheme

( ) ( ) ( ) ( ) ( ) 1 1 2 1 1 1 , 2 1 2 , 2 i i i i i i i i i t t t α α β β + + + + = + ∆ + - ∆ = + ∆ + + - δ δ δ δ δ δ δ δ δ ɺ ɺ ɺɺ ɺɺ ɺ ɺɺ ɺɺ (17)
where the parameters α=1/2 and β=1/4 define the constant average acceleration scheme. As Eq. ( 16) is non-linear, an iterative procedure is employed to treat the non-linearity. Namely, the linearization of Eq. ( 16) is performed by building a first-order Taylor series of this equation and the Newton-Raphson iterative strategy is taken into consideration as follows

( ) ( ) ( ) 1 1 1 1 1 , k i k k i i i d d + + + + + + = δ G G δ G δ ∆δ 0 δ ≃ (18) then ( ) ( ) 1 1 , NL k k i i + + = - J δ ∆δ G δ (19)
where

1 1 1
k k i i + + + = - ∆δ δ δ
is the iterative increment of the displacement vector and k is the Newton-Raphson iteration. The Jacobian matrix J NL is a function of the displacement vector

) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 2 1 1 1 , 1 , k i k k i i NL k i i i i be i i d t t d t t d t d α β β + + + + + + + + + = = + ∆ ∆ + - δ δ δ G J δ M C δ F K δ ɺ ( 20
)
where I is the identity matrix. The derivative of the non-linear hydrodynamic forces in the Jacobian matrix of Eq. ( 20) is a function of both the displacement and velocity vectors

( ) 1 1
,

k k i i + +
δ δ ɺ , updated at each iteration k and expressed by

( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 , , ,
.

k k k k k k i i i i i i be be be i i i d d t α β + + + + + + + + + ∂ ∂ = + ∂ ∆ ∂ δ δ δ δ δ δ F F F δ δ δ ɺ ɺ ɺ ɺ (21)
The procedure of the Newton-Raphson incrementaliterative strategy combined with the Newmark time integration scheme is presented as pseudo-code hereafter.

(1) Initial conditions 1

1 ; δ δ ɺ ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 0 0 , 0 0 - = + - - be δ M F F δ δ C δ K δ ɺɺ ɺ ɺ
(2) Prediction step: ( ) In order to reach the steady-state periodic regime, the Newmark algorithm has to be used over a large number of time periods. This results in an enormous computational time when the algorithm is repeated for a large number of fundamental excitation frequencies. Computational time can be saved by seeking directly the periodic solution by means of the shooting method being discussed in the next section.

1 1 1 1 1 1 1 1 1 2 ; , 1 k k k k i i i i k k i i t t α β β + + + + + + + + + = + = + ∆ = + ∆ δ δ ∆δ δ δ ∆δ δ δ ∆δ ɺ ɺ ɺɺ ɺɺ (

PERIODIC SOLUTION USING THE SHOOTING METHOD

The following two-point boundary-value problem defined by the periodicity condition is considered

( ) ( ) ( ) ( ) ( ) 0 , 0 , 0 , r r t t τ τ = = = - = H X X X X 0 (22) with , , T T T = X δ δ ɺ (23)
where τ r is the minimal period of the rotor response. The dependence of the system on the fundamental period τ ex of the periodic exciting forces is assumed, i.e., τ r =jτ ex (with j=1,2,3...). Integer multiples of the fundamental period are used to calculate sub-harmonic responses occurring, for example, after a period-doubling bifurcation. Practically, in order to find the minimal period τ r of the response, the shooting method is first performed with τ r =τ ex and if there is no convergence, it is then performed with τ r =2τ ex , τ r =3τ ex , ... successively. The state-space vector X in Eq. ( 22) is the solution of Eq. ( 5) at the end of one period τ r for an approached initial solution X(0). This vector is computed by the non-linear Newmark time integration procedure over one period τ r (see Section 5). A consistent linearization of Eq. ( 22) is made by finding a first-order Taylor series expansion of this equation and a Newton-Raphson iterative correction is executed

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 , 0 , 0 , , 0 k r k k r r τ τ τ + ∂ + = ∂ X H H X H X ∆X 0 X ≃ (24) then ( ) ( ) ( ) ( ) 0 , 0 , , k k SH r r τ τ = - J X ∆X H X ( 25
)
where ∆X = X k+1 (0) -X k (0) represents the iterative increment of the state-space vector and k is the Newton-Raphson iteration. The Jacobian matrix SH J is a function of the state-space vector X(0) at iteration k and defined by

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 , 0 , 0 , , 0 0 k k r r k SH r τ τ τ ∂ ∂ = = - ∂ ∂ X X H X J X I X X (26) with ( ) ( ) ( ) ( ) ( ) ( ) 0 ,
, 0 k r r r r τ τ τ τ   ∂ = =   ∂   X Φ X φ Φ X ɺ (27)
where φ(τ r ) is the monodromy matrix. The differentiation of Eq. ( 5) with respect to the vector X(0) yields

( ) ( ) ( )

. be be

t t t ∂ ∂     + + + + =     ∂ ∂     F F M Φ C Φ K Φ 0 δ δ ɺɺ ɺ ɺ (28)
Equation (28) is a system of linear differential equations. In order to solve it and to obtain the matrices ( )

r τ Φ and ( ) r τ Φ ɺ
forming the monodromy matrix φ(τ r ), the linear version of the Newmark algorithm is used over one period τ r of the rotor response.

The shooting algorithm solving simultaneously Eqs. ( 5) and ( 22) is presented as pseudo-code hereafter.

(1) Approached initial solution 1 X(0) (2) Calculation of the state-space vector X (3) Residual calculation H with Eq. (22) (4) Calculation of the Jacobian matrix J SH with Eq. (26) (5) Newton-Raphson increment ∆X with Eq. (25

) (6) Correction step X k+1 (0)=X k (0)+∆X (7) If iterations SH l >
, increase the period of the rotor response and return to (1)

(8) If SH ε > H X
, return to (2) for a new Newton-Raphson iteration, else stop the algorithm After the periodic solution is obtained, its stability can be verified by applying the Floquet theory, i.e., by calculating the eigenvalues of the monodromy matrix called "Floquet multipliers". The solution is asymptotically stable if every Floquet multiplier remains inside a unit circle in the complex plane. When any Floquet multiplier leaves the unit circle, the periodic solution losses its stability and a bifurcation begins to appear. In practice, the saddle-node or period-doubling bifurcations occur if any Floquet multiplier leaves through +1 or through -1 respectively, while the quasi-periodic bifurcation occurs if two complex conjugate Floquet multipliers leave transversely the unit circle.

NUMERICAL RESULTS AND DISCUSSION

Configuration and Data

The symmetric rotor-hydrodynamic bearing system illustrated in Fig. 2 is excited by both a rotating mass unbalance and a sinusoidal translation of the base along the Oz axis expressed as

z O =Z O cos(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 with eight identical 2-node Timoshenko beam finite elements i.e., the total number of degrees of freedom is n g =36. The disk is located at node 5 and the bearings # 1 and # 2 are placed at nodes 1 and 9 respectively, i.e., the corresponding localized non-linear degrees of freedom are 1-4 and 33-36. The rotor runs at a speed of rotation Ω=1200 rpm (=20 Hz, the mass unbalance frequency), the mass unbalance being located on the disk.

The relative coordinates of the static equilibrium position of the shaft geometric center in the fluid film bearings are given by δ be 0 /c be =<-0.29,-0.88> R T .

The overall dynamic analysis of the non-linear on-board rotor is done by means of bifurcation diagrams, rotor orbits and Poincaré maps.

Bifurcation Diagrams

In the case of a harmonic external excitation of period τ ex , a linear rotor system has always a harmonic response of same period, i.e., τ r =τ ex , while a non-linear rotor system can have a non-periodic response or a periodic response with different period due to bifurcations. Such changes in the non-linear dynamic regime of the system can be identified by means of bifurcation diagrams.

Figure 3 shows the bifurcation diagram of the on-board rotor at the bearing # 2 (node 9) excited by the mass unbalance (m mu r mu =1500 g mm) and the sinusoidal translational motion of the base of frequency Ω z =80 Hz.

In the diagram, the amplitude Z O of the base motion is used as a bifurcation parameter to provide essential information about the Poincaré sections. The step used for this parameter Z O is 0.05×10 -5 m. Since the excitations due to the mass unbalance and to the base motions are sinusoidal and commensurable, . The initial conditions for the simulations are set to δ be 0 /c be . For 1×10 -5 ≤Z O ≤4.95×10 -5 m, the rotor motion is harmonic of period τ r =τ ex because there is only one point obtained by the Poincaré sections. Then, the first bifurcation appears and the rotor vibration displays a period-doubling orbit with τ r =2τ ex when 5×10 -5 ≤Z O ≤5.1×10 -5 m. At Z O =5.15×10 -5 m, the rotor follows a quasi-periodic orbit, but it returns to the harmonic motion for 5.35×10 -5 ≤Z O ≤6.3×10 -5 m. For the rest of the diagram, the solution of the equations of motion undergoes a series of bifurcations as shown in Fig. 3.

In order to visualize successive period-doubling bifurcations which lead to quasi-periodic responses and to chaos, a zoom of the bifurcation diagram is reproduced in Fig. 4 for the zone of interest The finer grid allows a better observation of the changes of nonlinear dynamic regime when compared with Fig. 3 for the same zone of interest. For example, the rotor motion holds 4τ experiodic and 5τ ex -periodic orbits (sub-harmonic) for Z O =5.17×10 -5 m and 5.26×10 -5 m respectively.

The stability of some periodic solutions obtained in Fig. 4 is predicted using the Floquet theory. The monodromy matrix is calculated at the end of one period τ r of the rotor response and its corresponding Floquet multipliers are presented in Fig. 5 within a unit circle in the complex plane. As can be seen, the solution is stable at the point A of Fig. 4 where Z O =4.95×10 -5 m with τ r =τ ex (see Fig. 5(a)) and also at C where Z O =5×10 -5 m with τ r =2τ ex (see Fig. 5(c)), while it is unstable and a period-doubling bifurcation occurs at B where Z O =4.96×10 -5 m with τ r =τ ex (see Fig. 5(b)) because one of the Floquet multipliers leaves the circle through the negative real axis at -1. This can be verified with the orbit of the rotor at the bearing # 2 which is 1τ experiodic (see Fig. 6

Bifurcation Responses of the Rotor-Bearing System

In this section, some orbits of the rotor and their projected Poincaré maps at the bearing # 2 (node 9) corresponding to values of interest of the bifurcation parameter Z O of the bifurcation diagram in Fig. 3 are presented.

All the orbits are made dimensionless with respect to the bearing clearance c be and are thus plotted within a unit circle (red curve).

The relative coordinates of the static equilibrium position of the shaft center in the bearings δ be 0 /c be are used as initial conditions for the transient dynamic calculations. The Poincaré maps are obtained by taking successive intersections of the relative dynamic displacements of the rotor with the instants corresponding to the multiples of the period τ ex =0.05 s of the harmonic external excitation of the rotor starting from t=501τ ex to eliminate the transient effects.

When the amplitudes Z O of the base translation are consecutively equal to 7×10 -5 m (Fig. 7

CONCLUDING REMARKS

A finite element model based on the Timoshenko beam theory is presented to analyze the dynamic behavior of a symmetric on-board rotor-non-linear hydrodynamic bearing system whose base is subject to sinusoidal translations.

The derivation of the equations of motion shows that the base rotations create time-varying parametric coefficients, while the base translations contribute only to the external force vector.

Robust Newmark integration and shooting algorithms combined with the Newton-Raphson iterative procedure are used to obtain the dynamic responses of the on-board rotor.

It is noted that depending on the amplitudes of the sinusoidal motions of the base, non-linear phenomena, such as perioddoubling and quasi-periodic motions and transition to chaos, are observed by means of bifurcation diagrams and rotor responses. The stability of periodic solutions is verified using the Floquet theory.

Figure 1 .

 1 Figure 1. FRAMES OF REFERENCE FOR THE ON-BOARD ROTOR

δ

  at iteration k and expressed by

(

8 )F δ δ ɺ ( 9 )

 89 Return to (3) for new hydrodynamic forces ( ) , be Return to (4) for a new residual G (10) If NL ε > G F , return to (5) for a new Newton-Raphson iteration, else return to (2) for a new time step

Figure 2 .

 2 Figure 2. ON-BOARD ROTOR-HYDRODYNAMIC BEARING SYSTEM CONFIGURATIONTable 1. ROTOR AND BEARING DATA Disk material density Disk radius Disk thickness Disk location Shaft material density Young's modulus of the shaft Poisson's ratio of the shaft Shaft radius Shaft length Mass unbalance

  time, the step of the bifurcation parameter Z O is 0.01×10 -5 m.

  (a)) for the point A and becomes 2τ experiodic (see Fig. 6(b)) for the point C.

Figure 4 . 3 FOR

 43 Figure 4. ZOOM ON THE BIFURCATION DIAGRAM IN FIG. 3 FOR THE ZONE OF INTEREST

Figure 5 .Figure 6 .

 56 Figure 5. EIGENVALUES OF THE MONODROMY MATRIX CORRESPONDING TO THE POINTS A,B AND C IN FIG. 4 ASSOCIATED WITH: ZO= (a) 4.95×10 -5 m, (b) 4.96×10 -5 m, (c) 5×10 -5 m, RESPECTIVELY

  (a)), 9×10 -5 m (Fig.7(b)) and 9.45×10 -5 m (Fig.7(c)), the rotor displacements exhibit a chaotic orbit, a quasi-periodic orbit and a subharmonic orbit (8τ ex -periodic) respectively.

Figure 7 .

 7 Figure 7. ORBITS AND POINCARÉ MAPS OF THE ROTOR AT THE BEARING # 2 DUE TO SINUSOIDAL TRANSLATIONS OF THE BASE: zO=ZOcosΩ z t WITH Ω z =80 Hz AND ZO= (a) 7×10 -5 m, (b) 9×10 -5 m, (c) 9.45×10 -5 m

  ,

	t	r W d sh	mu	t	mu b	t	d sh b	t	c d sh b	t	t
		s d sh b	t	t


Marc Sansa

Nghi Van

Sébastien Nguyen

Claude-Henri Baguet

Régis Lamarque

Sébastien Dufour

Hentz

Real time sensing in the non linear regime of NEMs resonators

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


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

Philippe Velex

INFLUENCE OF THE NONLINEAR DYNAMIC BEHAVIOR OF JOURNAL BEARINGS ON GEAR-BEARING ASSEMBLIES

Journal bearings cannot be considered as passive elements in gear-bearing assemblies, and the lubricant is recognized as playing an important role in the interactions between the shafts and the bearings. In order to take this influence into account, bearings are usually modeled by means of eight dynamic coefficients, i.e., asymmetric stiffness and damping matrices.

In this paper, a nonlinear approach is proposed enabling the behavior of a gear-shaft-bearing assembly to be analyzed. A discrete finite element model is used for the shafts, and a specific gear element is introduced which accounts for non-linear time-varying mesh stiffness as well as tooth shape deviations. The meshing forces are internal system forces whereas the effects of the bearings on the shafts are taken to be external. A combination of the Newmark time integration scheme and the Newton-Raphson algorithm is used to simultaneously solve the contact problem for the gear, and the Reynolds equations for the bearings. The resulting algorithm is applied to a single stage geared system with two shafts, four bearings, a pinion and a gear while taking mass unbalance, eccentricity and meshing excitations into account. Several examples are presented which demonstrate the influence of bearing nonlinearity and the efficiency of the proposed model and numerical procedure.

INTRODUCTION

The dynamic response of geared systems remains a major concern in terms of noise and vibration, especially in highspeed applications. In this context, journal bearings are recognized as interesting technological solutions since they provide significant stiffness and damping for reduced noise levels compared with rolling element bearings. The literature on journal bearings comprises numerous contributions, only a few of which can be cited here [START_REF] Holmes | The Vibration of a Rigid Shaft in Short Sleeve Bearings[END_REF][START_REF] Dowson | A Generalized Reynolds Equation for Film Fluid Lubrication[END_REF][START_REF] Lund | The Stability of an Elastic Rotor in Journal Bearings with Flexible Damped Supports[END_REF][START_REF] Booker | Dynamically Loaded Journal Bearings: Mobility Method of Solution[END_REF][START_REF] Rhode | A Generalized Short-Bearing Theory[END_REF][START_REF] Ferron | A Study of the Thermodynamic Performance of a Plain Journal Bearing -Comparisons between Theory and Experiments[END_REF][START_REF] Fantino | Comparison of Dynamic Behavior of Elastic Connecting Rod Bearing in Both Petrol and Diesel Engines[END_REF] and it is now accepted that, for most practical cases, the theoretical foundations are firmly established allowing accurate predictions of bearing performance. A survey about the theory of hydrodynamics and its applications to various bearing geometries and operating conditions can be found in the classic textbooks of Cameron [START_REF] Cameron | The Principles of Lubrication[END_REF], Pinkus and Sternlicht [START_REF] Pinkus | Theory of Hydrodynamic Lubrication[END_REF], Frêne et al. [START_REF] Frene | Hydrodynamic Lubrication -Bearings and Thrust Bearings[END_REF].

On the other hand, the simulation of gear dynamics has yielded a vast literature [START_REF] Özgüven | Mathematical models used in gear dynamics. A review[END_REF], [START_REF] Velex | Modélisation du comportement dynamique des transmissions par engrenages', Chapter 2 of 'Comportement dynamique et acoustique des transmissions par engrenages[END_REF] with the majority of the gear models based upon lumped parameter representations combining rigid gears, discrete elastic and dissipative elements [START_REF] Blankenship | A comparative study of selected mesh force interface dynamic models[END_REF], [START_REF] Blankenship | A new gear interface dynamic model to predict multi-dimensional force coupling and excitation[END_REF], [START_REF] Gregory | Dynamic behaviour of spur gears[END_REF] or, more recently, on combinations of shaft finite elements and elastic foundations [START_REF] Ajmi | A model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears[END_REF]. Forcing terms are often derived from transmission errors (Gregory et al. [START_REF] Gregory | Dynamic behaviour of spur gears[END_REF], Munro [START_REF] Munro | The dynamic behaviour of spur gears[END_REF], Özgüven and Houser [START_REF] Özgüven | Dynamic analysis of high speed gears by using static loaded transmission error[END_REF]) and /or the contact conditions between the active flanks [START_REF] Velex | A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour[END_REF]. Non linear features in spur gearing such as backlash and shocks on tooth flanks have been analyzed by Kahraman and Singh [START_REF] Kahraman | Non-linear dynamics of a spur gear pair[END_REF], Kahraman and Blankenship [START_REF] Kahraman | Experiments on non-linear dynamic behavior of an oscillator with clearance and periodically time-varying parameters[END_REF], Parker et al. [START_REF] Parker | Modeling the non linear vibration of a spur gear pair[END_REF].

Over the last decade, computing procedures have been developed which include the characteristics of shafts, gears, bearings and housings in the prediction of the global vibrational behavior of a transmission [START_REF] Baud | Static and dynamic tooth loading in spur and helical geared systems. Experiments and code validation[END_REF][START_REF] Houpert | Prediction of bearing, gear and housing performances[END_REF][START_REF] Bourdon | Static rolling bearing models in a C.A.D. Environment for the study of complex mechanisms: Part I -Rolling bearing model[END_REF][START_REF] Bourdon | Static rolling bearing models in a C.A.D. Environment for the study of complex mechanisms: Part II -Complete Assembly model[END_REF]. The majority of these codes, however, are restricted to linear analysis, i.e., static solution and modal analysis, and bearing and gear models remain global. The present work contributes to the analysis of the interactions between gears, shafts and journal bearings in geared drives. In contrast to the majority of the models in the literature, the time-varying properties and non-linearities of bearings and gears are introduced in the simulations. To this end, the equations of motion, the normal contact problems on tooth flanks and the Reynolds equation for hydrodynamic bearings are simultaneously solved by combining a time-step integration scheme, a Newton-Raphson procedure and a normal contact algorithm. Finally, the importance of gear-bearing dynamic interactions and the interest of non-linear models for journal bearings are discussed. C : radial clearance ε : eccentricity ratio (varies between 0 and 1) φ : attitude angle (rad)

NOMENCLATURE

-DESCRIPTION OF THE MODEL 1-Shaft-gear model

The dynamic model of single stage transmissions (figure 1) is based on the developments of Velex and Maatar [START_REF] Velex | A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour[END_REF] and Ajmi and Velex [START_REF] Ajmi | A model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears[END_REF]. Shafts are modeled by classic two-node finite elements with 6 degrees of freedom per node for bending (Timoschenko's beam), torsion and axial displacements. A pinion -gear pair are represented by two 2-node shaft elements connected by lumped stiffnesses which account for contact, tooth and gear body deflections along the contact lines in the base plane (figure 2). An elemental mesh stiffness and an equivalent normal shape deviation are associated with every potential point of contact to simulate actual tooth flank traces and their evolutions with time. Based upon rigid-body kinematics, the lines of contact are translated and all relevant parameters (stiffness and deviations) are re-calculated at each time step of the meshing process. It is assumed that the contacts between mating flanks are line contacts within the theoretical base plane and that tooth friction forces can be neglected compared with normal forces. The normal contact condition at every potential point of contact on the tooth flanks is directly included in the mesh stiffness formulation by canceling the elemental stiffness when the corresponding deflection is negative or nil. The proposed formulation leads to time-varying non-linear mesh stiffness matrices and forcing terms, the latter accounting for tooth shape deviations and mounting errors [START_REF] Ajmi | A model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears[END_REF], [START_REF] Velex | A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour[END_REF], [START_REF] Baud | Static and dynamic tooth loading in spur and helical geared systems. Experiments and code validation[END_REF]. The pinion-gear mass matrix is classic since second order terms and gyroscopic components are neglected. or its local (or polar) co-ordinates in the frame ) , ( q r r r

) , ( r x r e O O r r r = = ′ φ (2)
Introducing the angular co-ordinate θ whose origin is on the axis r r

, the lubricant film thickness is given as:

) cos 1 ( θ ε + = C h ( 3
)
where the eccentricity ratio C e / = ε varies between 0 and 1. Assuming an isothermal, laminar flow of an iso-viscous incompressible fluid, the behavior of the lubricant film is governed by the Reynolds equation [START_REF] Dowson | A Generalized Reynolds Equation for Film Fluid Lubrication[END_REF]. For the sake of simplicity, the short bearing approximation is introduced, i.e., the width-to-diameter ratio D L / is supposed to be small, which leads to the following Reynolds equation for dynamic conditions: 4) can be integrated and the hydrodynamic pressure field is expressed as:

      + - =       ∂ ∂ ∂ ∂ θ ε θ φ ω µ cos 2 ) 2 ( 6 3 & & C d dh z p h z (4) Using the boundary conditions 0 ) 2 / , ( = ± = L z p θ , Equation (
( ) 3 2 2 2 ) cos 1 ( cos 2 sin ) 2 ( 4 3 , θ ε θ ε θ ε φ ω µ θ + - -         - - = & & L z C z p (5)
The hydrodynamic force b F is obtained by integration of ) , ( z

p θ for ] 2 , 0 [ π θ ∈ and ] 2 / , 2 / [ L L z - ∈
using the Gümbel boundary conditions which discard negative pressures from the force integral [START_REF] Frene | Hydrodynamic Lubrication -Bearings and Thrust Bearings[END_REF].

1-2-1 Steady-state solution

The solution for steady-state conditions can be deduced from ( 5) by setting the time-varying perturbations ε& and φ & to zero. The associated hydrodynamic bearing force integration is straightforward and gives

) 1 ( 16 ) 1 ( ) ( 2 0 2 2 0 2 2 0 0 2 2 0 ε π ε ε ε ω µ ε - + -             = C R D L LR F b (6)
with the attitude angle being defined as

0 2 0 0 1 4 tan ε ε π φ - = (7)
The steady-state position and bearing force depend on 0 ε . This is derived from the static equilibrium between the hydrodynamic reaction generated by the bearing and the external force 0 W imposed by the transmitted torques and transferred to the bearings by the pinion and the gear, i.e.,

0 ) ( W F b = ε ( 0
) 8
This equation is non-linear and it is solved by a dichotomy scheme. In turn, the eccentricity of the shaft in the journal modifies the pinion-gear center distance slightly and consequently, the gear mesh stiffness function and excitations.

1-2-1 Dynamic characteristics

Denoting ( ) 0 0 , y x , the shaft center co-ordinates in steady-state conditions and ( )

y x ∆ ∆ ,
, the corresponding timedependent dynamic perturbations, the actual position of the shaft center is defined as:

y y C x x C y y y x x x O O r r r r ) sin ( ) cos ( ) ( ) ( 0 0 0 0 0 0 ∆ + + ∆ + = ∆ + + ∆ + = ′ φ ε φ ε (9)
From which, the dynamic parameters ε ,φ , and their time- derivatives ε& and φ & are deduced as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y C x C x t y C x C C y y C x x C t y C x C x C C y C x C ∆ + ∆ + + ∆ - = ∂ ∂ = ∆ + + ∆ + ∆ ∆ + + ∆ ∆ + = ∂ ∂ = ∆ + + ∆ + ∆ + = ∆ + + ∆ + =
φ ε ε ε φ ε ε φ φ φ ε φ ε φ ε φ ε ε ε φ ε φ ε φ ε φ φ ε φ ε ε (10)

-DYNAMIC BEHAVIOR OF THE GEAR-BEARING ASSEMBLY 2-1 Equations of motion

The assembly of all the elemental matrices and forcing term vectors leads to the following system of equations in the frame ) , , ( z t s r r r relative to the structure

)} , ( { )} , ( { } )]{ , ( [ } ]{ [ } ]{ [ X X F X t F X X t K X C X
          =           = 0 ) sin( ) cos( 0 2 2 t Me t Me F F F mi mi miY miX mi ω ω ω ω (12)
for each node of the shaft. In this expression, M is the mass of the shaft plus the pinion or gear and mi e represents the distance between the inertia and geometrical centers of the shaft.

In the proposed formulation [START_REF] Özgüven | Mathematical models used in gear dynamics. A review[END_REF], the contributions of bearings appear as external forces to the system following the original approach of Abdul-Wahed [START_REF]Comportement Dynamique des Paliers Fluides -Etude Linéaire et Non-Linéaire[END_REF] in his study of the nonlinear behavior of rigid shafts supported by hydrodynamic bearings. The system of equations ( 11) is non-linear because the hydrodynamic forces ) , ( X X F b & depend on the position X and on the velocity X & , and because of the contact conditions between the gear teeth. The resolution requires a complex procedure combining several iterative schemes and a time-step integration process as described in Section 3 -3, which will be referred to as non-linear analysis. The solution technique can be significantly simplified by assuming small displacements in the vicinity of the static positions, i.e., using the linear analysis which is probably no longer suited to the case of significant dynamic loads or mass imbalances. Both kinds of analyses, linear and non-linear, are detailed in the following section.

2-2 Linear analysis

Using a first order expansion in the vicinity of the static solution 0 X leads to the following dynamic bearing force vectors:

{ } { } X C X K F X X F XY XY b b & & ∆ - ∆ - = ] [ ] [ } { ) , ( 0 ( 13
)
where for bearings on the output (gear) shaft , with the attitude angle 0 φ obtained by (7).

0 b F is obtained by (6), 0 X X X - = ∆ , XY K ] [
) ( 4 ) ( 1 ) 2 1 ( ) ( 1 ) ( 1 ) 1 ( 8 ] [ ] [ 2 0 2 2 0 0 0 0 2 0 0 2 0 0 2 0 0 0 2 0 2 0 0 0 ε π ε ε ε ε ε ε ε π ε ε ε π ε ε ε ω - + = - = = = - = = - + - = - = - + =       =       = f A B B A B A B f A f A f A f A B B B B C W C A A A A C W K
Finally, the matrices

1 0 0 1 0 0 ] [ )] ( ][ [ )] ( [ ] [ )] ( ][ [ )] ( [ - - = = Q C Q C Q K Q K xy XY xy XY ε ε ε ε (18) with       - - - = α α α α sin cos cos sin ] [Q (19)
for all bearings, where α is the pressure angle.

Replacing the expression ( 13

) of ) , ( X X F b
& into the equations of motion [START_REF] Özgüven | Mathematical models used in gear dynamics. A review[END_REF] along with the expressions of X ∆ and

X & ∆ , one obtains the linear system of equations in the frame ) , , ( z t s r r r

} { )] ( [ } { )} , ( { } ]{ [ } ]{ [ } ]{ [ 0 0 0 0 0 X K F X t F X K X C X M XY b ε + + = + + & & & (20)
where

XY gear shaft XY K K K K C C C )] ( [ ] [ ] [ ] [ )] ( [ ] [ ] [ 0 0 0 0 ε ε + + = + = (21)
The differential system (20) is solved by the implicit Newmark time-step integration scheme combined with a normal contact algorithm which verifies that all contact forces on the teeth are positive and no contact deflections occur outside the contact area [START_REF] Velex | A mathematical model for analyzing the influence of shape deviations and mounting errors on gear dynamic behaviour[END_REF].

Table 1 -Iterative procedure

2-3 Non-linear analysis

Because of the assumption of small perturbations, the linear theory cannot account for large dynamic loads. Solving the complete non-linear problem requires the Reynolds equation in dynamic regime, the contact conditions on the teeth and the equations of motion to be dealt with simultaneously. In this case, the hydrodynamic forces

) ( 1 , 1 k n n X t K + + and ) ( 1 , 1 k n n X t F + + Evaluate the hydrodynamic forces ) , ( 1 1 k n k n b X X F + + & Evaluate the residual vector ) ( 1 k n X G + Compute the jacobian matrix ) ( 1 k n X J + Compute the correction increment ) ( ) ( 1 1 k n k n X G X X J + + - = ∆ Evaluate 1 1 + + k n X , 1 1 + + k n X & and 1 1 + + k n X & & Compute the dynamic deflection for ∈ i M contact line Evaluate the residual vector ) ( 1 1 + + k n X G
Is there i M with pressure < 0 ? Is there i M with a positive deflection ? [Q are the matrices in ( 16), ( 17) and ( 19)

l X G k n ≤ + + ) ( 1 1 ? n n n n n n X X X X X           = = ∫∫ ∫∫ dz Rd z p dz Rd z p F F F sin ) , ( cos ) , ( (22)
      =             =      
The non-linear equations ( 11) are solved by combining the Newmark time integration scheme with the Newton-Raphson incremental-iterative algorithm, and with an iterative process aimed at updating the dynamic characteristics of the meshing process (Table 1).

-NUMERICAL RESULTS

The gear, shaft and bearing data used in the numerical simulations are given in Tables 2 to 4. A unique modal damping factor of 0.03 has been used for all the numerical simulations and the time increment t ∆ was set to 1/32th of the mesh period. All simulations were performed over 256 mesh periods, i.e. 8192 time increments, which correspond approximately to 10 pinion revolutions.

-1 Gear-bearing interactions

A first simulation was performed with an input (pinion) speed ω=700 rad/s, an input torque C =200N.m and the mass imbalance forces in [START_REF] Velex | Modélisation du comportement dynamique des transmissions par engrenages', Chapter 2 of 'Comportement dynamique et acoustique des transmissions par engrenages[END_REF] were computed with mi e =0.5mm. The corresponding transient motion of the shaft center in bearing 1 (input shaft) is plotted in Figure 4a. The motion, composed of 10 loops corresponding to the 10 pinion revolutions, becomes stable after the transient oscillations on the first orbits have disappeared. The last calculated orbit is shown in Figure 4b which shows the 26 oscillations associated with the mesh period and demonstrates the interaction between the gears and the bearings. The norm of the hydrodynamic forces over the last orbit, i.e. the last 256 mesh periods, of the same bearing is plotted in Figure 5. The low-frequency excitation seems prevalent but, as in the previous example, the mesh frequency components appear in the time signal superimposed on the basic once-per-revolution period related to mass imbalance.

-2 Comparison of the linear and the non-linear theories

Dynamic tooth loading is characterized by considering the dynamic tooth load factor R defined as the ratio of the maximum dynamic mesh force to the maximum static one.

Figure 6 represents the evolutions of R versus the pinion speed calculated for various loads and using different solution techniques. The solutions obtained by the linear and non-linear methods are very close and are dominated by amplitude jumps and shocks at the major tooth critical frequency. It can also be observed that dynamic amplifications are reduced when the transmitted torque is increased. Considering bearing forces, the results on the pinion shaft (Figure 7) reveal that, here again, the linear and non-linear theories lead to very similar response curves with slightly larger differences as the transmitted load increases. The situation is different on the gear shaft, as illustrated in Figure 8, for which the non-linear and linear results deviate significantly as either speed or load increases. This discrepancy is apparently caused by the larger imbalance on the gear. This observation is confirmed by the bearing force plots in Figure 9, obtained for three imbalance amplitudes ( mi e =0mm, mi e =0.5mm and mi e =1mm), which indicate that larger dynamic forces make non-linear modeling necessary. On the other hand, dynamic tooth loads appear as moderately sensitive to imbalance, as illustrated in Figure 10, and, in contrast to bearing forces, the linear and non-linear results remain very close whatever the imbalance amplitude. The final set of results in Figures 11 and12, shows, for two imbalance amplitudes ( mi e =0.5mm and mi e =1mm), the trajectories of the pinion shaft in its bearings for a pinion speed ranging from 100 to 1200 rad/s. The previous conclusions about the role of dynamic force amplitudes on shaft-bearing interactions are confirmed and suggest that linear theories might not be suited for accurate vibration transfer analysis.

CONCLUSION

This paper presents a model of gear-shaft-bearing systems which accounts for various contact non-linearities and couplings. The state equations are solved by an original technique which combines a Newmark scheme, a Newton-Raphson procedure for bearing non-linearities, and a unilateral contact algorithm which deals with the contact problem on gear teeth. The proposed theory is applied to a simplified F.E. model of a single-stage spur gear system with 2 shafts and 4 bearings.

From the numerical examples, the following observations can be made: i) for accurate predictions of vibration transfer through the bearings, bearing non-linear behavior has to be taken into account in the case of large dynamic forces, ii) mesh excitations appear in the dynamic bearing response although bearing force amplitudes are usually dominated by imbalance, iii) dynamic tooth load sensitivity to once-per-revolution excitations seems very limited.

The present study is part of a research program on dynamic couplings in geared systems and is clearly approximate in nature since it is based on the short-bearing theory which limits the practical range of applications. Further work is under way to account for the influence of helical gears, different bearing technologies and flexible mounts.

:

  polar local frame of the bearing ω : rotation speed (ras/s)

Figure 1 -

 1 Figure 1 -Sketch of the geared system

Figure 2 -

 2 Figure 2 -Model of mesh elasticity and tooth shape deviations

Figure 3 - 2 1-2 Journal bearing model 1 - 2 - 1

 32121 Figure 3 -Direction of external load and definition of local frames for the bearings on shaft 1 and 2

  the stiffness and damping matrices due to the lubricant film.For short bearings, and applying Gümbel's conditions, the expressions of ] [K and ][C are easier to obtain in the the bearing. According to Frêne et al.

W

  is the static load generated by the meshing and applied to the bearing. The coefficients ij A and ij B depend on 0 ε only, which characterizes the steady-state position,

Figure 4 -Figure 5 -Figure 6 -

 456 Figure 4 -a) Transient motion of the shaft center of bearing 1 and b) last calculated orbit

Figure 7 -Figure 8 -Figure 9 -Figure 10 -Figure 11 - 1 -Figure 12 -

 7891011112 Figure 7 -Peak-to-peak of the hydrodynamic load b F in

  MM and C are the global constant mass and damping matrices

	X	,	X & ,	X & &	are the degree-of-freedom, velocity and acceleration
	vectors
	K	, ( X t	)		is the global stiffness matrix of the system,
	K	( t	,	X	)	=	K	shaft +	K	gear	( t	,	X	)	with
								K		shaft		: constant global stiffness matrix of the shafts
								K gear	, ( X t	)	: global stiffness matrix of the gear which is
						time-dependant because of the contact length evolution on
						the base plane and potentially non-linear because of the
						interaction between the degrees of freedom, tooth
						deviations and the instantaneous contact extent (Velex and
	F	( t	Maatar, [19]) ) ( ) , 0 t F X =	+	F gear	( t	,	X	)	with
	( 0 t F	)	: external load vector (input and output torques, mass
	unbalance, …)
	F gear		, ( X t	)	: embodies the inertial effects caused by unsteady
	rotations (associated with no-load transmission error) and the
	contributions of geometrical errors and tooth shape deviations
	(pitch errors, tooth modifications …) [19]
	F b	) & are the hydrodynamic forces produced by the ( X , X
	bearings which are opposing the motions of the shaft as
	described in section 2.
										If the inertia center of the shaft is different from its
	geometrical center, then mass imbalance forces are embedded in the external load vector ) ( 0 t F . In the frame ) , , ( z t s r r r , these
	forces read

Table 2 -

 2 Gear data

		Pinion	Gear
	Number of teeth	26	157
	Face width (mm)	50	40
	Helix angle (°)	0
	Module (mm)	4
	Pressure angle α (°)	20
	Dedundum coefficient	1.	1.
	Addendum coefficient	1.4	1.4
	Profile shift coefficient	0.16	-0.16
	Center distance (mm)	366
	Mass (kg)	2	74
		Pinion shaft	Gear shaft
	Outer diameter (mm)	70	90
	Inner diameter (mm)	30	30
	Shaft length (mm)	1 L = 2 L =160
	Young modulus (N/m 2 )	2.1.10 11
	Density (kg/m 3 )	7800

Table 3 -

 3 Shaft data

		Bearings 1 and 2	Bearings 3 and 4
		Pinion shaft	(gear shaft)
	Diameter (mm)	70	90
	Length (mm)	14	18
	Radial clearance (µm)	150	110

Table 4 -

 4 Bearing data

& & & & & &

& & & &

& & & & & &

where